Molar mass of AgF = 126.8666032 g/mol. The molar mass is used to convert between grams and moles and the unit is g/mol. The formula weight is simply the weight in atomic mass units of all the atoms in a given formula. To complete this calculation, you have to know what substance you are trying to convert.

Directions: Use dimensional analysis to …

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Molar mass of AgNO3 is 169.873, 2 kg AgNO3 is equal to how many moles?

The molar mass of an element is equal to its... Molar Mass DRAFT.

What is the mass of 2.23 x 10 23 atoms of sulfur?

Molar mass of AgF = 126.8666032 g/mol.

This compound is also known as Silver Fluoride.

The molar mass is the amount of grams in a mole of a substance.

Molar mass can then be used as a conversion factor to convert amounts in moles to amounts in grams.

For example, if you have 1.5 mol of NaCl dissolved in 0.500 L …

The atomic weights used on this site come from NIST, the National Institute of Standards and Technology.

The molar mass of Silver is about 107.86820 grams per mole.

Molar mass of AgF = 126.8666032 g/mol.

The formula weight computed is the molecular weight.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

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Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

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Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol.

Molar mass of AgF = 126.8666032 g/mol


Recursion is a problem-solving approach that involves breaking down complex problems into smaller, manageable parts. Technically, it is "the process in which a function calls itself directly or indirectly," and a function that calls itself is referred to as a **recursive function**. This concept can be invaluable in solving programming problems.

To illustrate recursion, consider a famous search algorithm. Imagine searching for a specific last name, "Patrick," in a stack of graded papers sorted alphabetically by last name. One approach is a **linear search**, starting from the beginning of the stack and flipping through each paper until finding "Patrick." However, a more efficient approach is **binary search**, which involves flipping to the middle of the stack and determining if the target name is in the top or bottom half. This process can be repeated, cutting the search space in half with each iteration.

For example, starting with 100 papers, two iterations of binary search can narrow the search space down to 25 papers by examining only two papers. This algorithm is a powerful example of recursion.

For recursion to work, two conditions must be met:

1. The problem must have a known solution for a small enough input (e.g., finding the target name "Patrick").
2. The problem set must decrease in size with each recursive iteration (e.g., discarding half of the papers with each binary search).

It is essential to note that any problem solvable with recursion can also be solved without it. However, recursion often provides simple and elegant solutions to complex problems, making it easier to translate abstract algorithms into code.

Now that the concept of recursion is understood, it can be applied to solve complex problems one step at a time.


12/27/2008, 08:29 AM

Hello,

I've been assessing the progress made in extending tetration. Although I'm happy with my own work, I worry that it's due to a lack of critique. Previously, I couldn't write up my work on tetration and the Ackermann function, but that has changed. With many excellent minds working on the subject, it's surprising that no one has published a paper extending tetration to everyone's satisfaction. This forum has presented an idea that the blockage in extending tetration comes down to establishing the uniqueness of the solution. Is this really the only issue in proving the extension of tetration?

I am interested in this idea and would like to explore it further. The uniqueness of the solution seems to be a crucial aspect, but I am unsure if it is the only obstacle. I look forward to discussing this topic and gaining a deeper understanding of the challenges in extending tetration.


# 4.2 Markup

The process of taking a product’s cost and increasing it by some amount to arrive at a selling price is called **markup**. This process is critical to business success because every business must ensure that it does not lose money when it makes a sale. From the consumer perspective, the concept of markup helps you make sense of the prices that businesses charge for their products or services.

## The Components in a Selling Price

The selling price of a product is composed of three components: cost, expenses, and profit. The cost is the amount of money that the business must pay to purchase or manufacture the product. Expenses are the financial outlays involved in selling the product, such as wages, taxes, leases, equipment, electronics, insurance, utilities, fixtures, décor, and many more. Profit is the amount of money that remains after a business pays all of its costs and expenses.

The relationship between these components can be expressed as:

[latex]S = C + E + P[/latex]

Where:
[latex]S[/latex] is the selling price
[latex]C[/latex] is the cost
[latex]E[/latex] is the expenses
[latex]P[/latex] is the profit

## Calculating the Markup Dollars

The markup amount is the difference between the selling price and the cost. It represents the total dollars of the expenses and profits. The markup amount can be calculated as:

[latex]M = E + P[/latex]

Where:
[latex]M[/latex] is the markup amount
[latex]E[/latex] is the expenses
[latex]P[/latex] is the profit

The selling price can also be expressed as:

[latex]S = C + M[/latex]

Where:
[latex]S[/latex] is the selling price
[latex]C[/latex] is the cost
[latex]M[/latex] is the markup amount

## Calculating the Markup Percent

The markup rate on cost is the percentage rate by which the cost of the product needs to be increased to arrive at the selling price. It can be calculated as:

[latex]m_C = \frac{M}{C} \cdot 100\%[/latex]

Where:
[latex]m_C[/latex] is the markup rate on cost
[latex]M[/latex] is the markup amount
[latex]C[/latex] is the cost

The markup rate on selling price is the percentage of the selling price that remains available as list profits after the cost of the product is recovered. It can be calculated as:

[latex]m_S = \frac{M}{S} \cdot 100\%[/latex]

Where:
[latex]m_S[/latex] is the markup rate on selling price
[latex]M[/latex] is the markup amount
[latex]S[/latex] is the selling price

## Break-Even Pricing

The break-even selling price is the price at which the business covers its costs and expenses but realizes no profit or loss. It can be calculated as:

[latex]S_{BE} = C + E[/latex]

Where:
[latex]S_{BE}[/latex] is the break-even selling price
[latex]C[/latex] is the cost
[latex]E[/latex] is the expenses

The break-even selling price is important because it helps businesses determine the minimum price they need to charge to cover their costs and expenses. If the selling price is below the break-even price, the business will incur a loss. If the selling price is above the break-even price, the business will make a profit.


Definite Integration Based on Property No 4 Problem No 1

To determine whether a function is even or odd is crucial in definite integrals with limits in the form of a to a. Even functions remain the same when the variable is replaced with its negative, while odd functions become the negative of the original.

Key properties of even and odd functions include:
- Even functions can be split from 0 to a and multiplied by 2 to simplify the integration process.
- The integral of an odd function over a symmetric interval is zero.
Checking for even or odd functions helps optimize calculations and exploit symmetry in definite integrals.

The process of checking evenness or oddness involves substituting x for -x in the function and comparing the result to the original. If the resulting function is equal to the original function, it is even. If the resulting function is the negative of the original, it is odd.

Questions & Answers
Q: What is the significance of checking whether a function is even or odd in definite integrals?
A: Checking if a function is even or odd helps determine how to split the integral and simplify the calculations.
Q: How is the property of evenness or oddness of a function determined?
A: To check if a function is even or odd, we substitute x for -x in the function. If the resulting function is equal to the original function, it is even. If the resulting function is the negative of the original, it is odd.
Q: What is the rule for splitting the integral if the function is even?
A: If the function is even, the integral can be split from 0 to a and then multiplied by 2.
Q: How is the integral adjusted if the function is odd?
A: If the function is odd, the integral is directly zero according to the given property.

The property of definite integrals focuses on determining if a given function is even or odd, especially when the limits of integration are in the form of a to a. If the function is even, the integral can be split from 0 to a and multiplied by 2. This simplifies the integration process by utilizing the symmetry of even functions.


# Percentage of Loss Calculator

When a business incurs a loss, understanding the extent of the loss is crucial. The percentage of loss is a useful metric for assessing the impact of losses and taking corrective actions. The formula for calculating the percentage of loss is:

Percentage of loss = (Loss / Cost Price) x 100

Here, the loss refers to the difference between the cost price and selling price of a product. If the selling price is lower than the cost price, the business incurs a loss.

Let's consider an example: A business buys a product for $100 and sells it for $80. To calculate the percentage of loss, first, find the loss:

Loss = Cost Price – Selling Price = $100 – $80 = $20

Then, calculate the percentage of loss:

Percentage of loss = (Loss / Cost Price) x 100 = ($20 / $100) x 100 = 20%

So, the business incurred a loss of 20% on the product.

Another example: A business buys a product for $500 and sells it for $550. To calculate the percentage of profit or loss:

Profit/Loss = Selling Price – Cost Price = $550 – $500 = $50

Percentage of profit or loss = (Profit/Loss / Cost Price) x 100 = ($50 / $500) x 100 = 10%

This indicates a profit of 10% on the product. Note that a negative percentage of profit or loss indicates a loss, while a positive percentage indicates a profit.

More examples:

1. A store bought T-shirts for $100 each and sold them for $80 each. The loss per T-shirt is $20.
Percentage of loss = (Loss / Cost Price) × 100% = (20 / 100) × 100% = 20%

2. A car owner sold his car for $20,000, which is 25% less than the original cost price. Let the original cost price be x.
Selling price = Original cost price – (25% of Original cost price)
$20,000 = x – (0.25x)
$20,000 = 0.75x
x = $20,000 / 0.75
x = $26,666.67

3. A company bought 500 shares at $100 per share and sold them for $80 per share. The loss per share is $20.
Total loss = Loss per share × Number of shares = $20 × 500 = $10,000
Total cost price = Cost price per share × Number of shares = $100 × 500 = $50,000
Percentage of loss = (Total loss / Total cost price) × 100% = ($10,000 / $50,000) × 100% = 20%

The percentage of loss is a valuable metric for businesses to assess losses and take corrective actions, such as adjusting prices, reducing costs, or finding new revenue streams. By knowing the percentage of loss, businesses can make informed decisions to mitigate losses and improve profitability.


Using MATLAB to find a generative equation for a sequence

The sequence is formed as 2*n*F_n + 1, where F_n is the n'th Fibonacci number. To find a generative equation for this sequence, we can use MATLAB to explore and derive a recurrence relation.

First, we generate the sequence using the Fibonacci numbers:
n = sym(0:100).';
Fn = fibonacci(n);
Sn = 2*n.*Fn + 1;

We can then examine the first few elements of the sequence:
Sn(1:10)

To derive a recurrence relation, we can use the null space of a Toeplitz matrix constructed from the sequence. We start by creating a Toeplitz matrix with the sequence:
X = toeplitz(Sn,[1,zeros(1,4)]);
rank(X(5:end,:))

If the resulting matrix is full rank, it means there is no linear combination of the columns that yields all zeros. We then try increasing the size of the Toeplitz matrix:
X = toeplitz(Sn,[1,zeros(1,5)]);
rank(X(6:end,:))

If the matrix is now rank deficient, it means there is a linear combination of the columns that yields all zeros. We can find this combination using the null space:
null(X(6:end,:))

The coefficients of the null space give us a recurrence relation for the sequence:
S(n+1) = 3*S(n) - S_(n-1) - 3*S(n-2) + S(n-3) + S(n-4)

We can test this recurrence relation by implementing it in MATLAB:
function Sn = Sfun(n)
switch n
case 0
Sn = 1;
case 1
Sn = 3;
case 2
Sn = 5;
case 3
Sn = 13;
case 4
Sn = 25;
otherwise
Sn = Sfun(n-5) + Sfun(n-4) - 3*Sfun(n-3) - Sfun(n-2) +3*Sfun(n-1);
end
end

We can then use this function to generate the sequence and compare it to the original sequence:
SfunM = memoize(@(N) Sfun(N));
SfunM(25)
2*25*fibonacci(sym(25)) + 1

The results match, confirming that the derived recurrence relation is correct.

This approach demonstrates how MATLAB can be used for explorational mathematics, deriving a generative equation for a sequence from its initial terms.


**Exploring Permutations**

We begin by defining key concepts:

- A **subgroup** of a group
- The **order** of an element of a group and what it means for an element to have **infinite order**
- An **isomorphism** between two groups and what it means for two groups to be **isomorphic**
- A **permutation** and the set of permutations for a set, denoted as , with the notation that permutations are written on the right

**Theorem 7**: For a set ,  is a group under composition, known as the **symmetry group** of . If , then  is non-Abelian, and . This theorem was verified by showing that  is non-Abelian through explicit examples of non-commuting permutations.

To count permutations, we consider the possibilities for where each element of the set is sent. Key definitions include:

- A **cycle** and its **length**
- A **transposition**
- **Disjoint cycles**, where two cycles do not share any elements

**Proposition 8**: Disjoint cycles commute. This was demonstrated by tracking where each element is sent under the composition of two disjoint cycles.

**Theorem 9**: Every permutation can be uniquely written as a product of disjoint cycles, up to cycling elements within cycles and permuting the cycle order. The existence part of this theorem was proven by considering the orbit of an element under a permutation, which forms a cycle, and then considering orbits of new elements until all elements are accounted for. This process terminates because there are only finitely many elements.

**Understanding and Application**:

- To deepen understanding, pick a permutation and decompose it into disjoint cycles using Theorem 9.
- Consider proving the uniqueness part of Theorem 9 independently.
- For further insight, explore the connection between bell-ringing and permutations.

**Preparation for Next Lecture**:

- Given a permutation as a product of disjoint cycles, determine its order and the necessary information for this calculation.
- Review the concept of permutation matrices from Linear Algebra II, focusing on determinants.
- Investigate how to express any cycle as a product of transpositions.

**Additional Insights**:

- Cycle types and how they change when a permutation is raised to a power can lead to interesting patterns, such as the apparent gain or loss of cycles. However, secret 1-cycles are always present, ensuring that cycles are not truly lost. Understanding when cycles break up or persist under repeated application of a permutation is a fascinating area of study.


**Boundary Conditions**

The objective of this lecture is to demonstrate how to determine the boundary conditions on the voltages and currents in a 2nd order circuit. These boundary conditions are used when calculating the transient response of the circuit.

**2nd Order Circuits**

A 2nd order differential equation is required to solve for the voltage across or the current flowing through a component. The circuit contains at least one resistor and the equivalent of two energy storage elements: 2 capacitors, 2 inductors, or a capacitor and an inductor.

**Boundary Conditions**

* Steady state: For step response functions u(t-t0) for all times between t = +/- except for some time period after t = t0.
* Capacitors are open circuits.
* Inductors are short circuits.
* During the transition at the step t = t0:
	+ Voltage across a capacitor is continuous: vC(t0+) = vC(t0-).
	+ Current through an inductor is continuous: iL(t0+) = iL(t0-).

**Initial Condition**

To find the initial condition, redraw the circuit at t < t0 and:
* Determine the value of all voltage and current sources at t < t0.
* Substitute an open circuit (resistor) for all capacitors: IC(t < t0) = 0A.
* Substitute a short circuit (0 resistor) for all inductors: VL(t < t0) = 0V.
* Calculate VC(t < t0) and IL(t < t0).

**Final Condition**

To find the final condition, redraw the circuit at t = ∞ and:
* Determine the value of all voltage and current sources at t = ∞.
* Substitute an open circuit (resistor) for all capacitors: iC(t = ∞) = 0A.
* Substitute a short circuit (0 resistor) for all inductors: vL(t = ∞) = 0V.
* Calculate vC(t = ∞) and iL(t = ∞).

**Examples**

Several examples are provided to illustrate the calculation of initial and final conditions for 2nd order circuits.

**Summary**

Calculation of the initial and final conditions for 2nd order circuits requires:
* Knowledge of the magnitude of the voltage and/or current sources in the circuit before and after a step function transition.
* In steady state (t < t0 and t = ∞), replace energy storage devices:
	+ Capacitors are open circuits => iC = 0A.
	+ Inductors are short circuits => vL = 0V.
* Calculate the voltage across the capacitor and the current through the inductor.
* During the transition at the step t = t0:
	+ Voltage across a capacitor is continuous: vC(t0+) = vC(t0-).
	+ Current through an inductor is continuous: iL(t0+) = iL(t0-).


#### CONTENTS

# EXCEL FUNCTIONS-COMBIN

## INTRODUCTION

The COMBIN function in Excel falls under the MATH AND TRIG category and returns the number of possible combinations with a specified number of items. This function is based on the combination concept in mathematics. To understand this concept, consider an example where we need to create a group of two persons from three different persons (A, B, and C). The possible groups are AB, AC, and BC, resulting in three groups.

The mathematical formula for this calculation is nCr, where n is the number of items and r is the size of the group. The formula is n! / (n-r)! * r!, where n! = n * (n-1) * (n-2) * ... * 1. Applying this formula to the example, we get 3! / (3-2)! * 2! = 3 * 2 / 2 = 3, which matches the manual calculation.

## PURPOSE OF COMBIN FUNCTION IN EXCEL

The COMBIN function returns the number of possible combinations with a specified number of items.

## PREREQUISITES TO LEARN COMBIN

To understand the COMBIN function, you should have:
- Basic understanding of using formulas or functions in Excel
- Knowledge of rows and columns in Excel
- Familiarity with financial terms (advantageous but not necessary)
- Excel software

## SYNTAX: COMBIN FUNCTION

The syntax for the COMBIN function is =COMBIN(number of items, number of items in subgroup), where:
- Number of items: total number of items
- Number of items in subgroup: number of items to be chosen from the total items

## EXAMPLE: COMBIN FUNCTION IN EXCEL

Consider two cases: one with 3 numbers and one with 4 numbers, selecting 2 numbers from each group. The analysis is done manually and using the COMBIN function.

### STEPS TO USE COMBIN FUNCTION

To demonstrate the function's usage, we take two examples with 3 and 4 numbers, forming groups of 2 numbers each. The function used is =COMBIN(3,2) and =COMBIN(4,2), which gives the same result as the manual calculation.

## KNOWLEDGE BYTES

### COMBINATION USING COMBIN FUNCTION

In mathematics, permutation and combination are related concepts. Permutation calculates the different possible arrangements of a given number of items, using the formula n! / (n-r)!. For combination, the formula is n! / (n-r)! * r!, as the order of selection does not matter (e.g., AB and BA are the same). The COMBIN function in Excel uses this combination formula to calculate the number of possible groups from a given number of items.


A matrix is a rectangular array of numbers, with its size or dimension defined by the number of rows and columns. Matrices can be classified into different types, and a matrix can belong to more than one type.

There are several types of matrices, including:
- **Row Matrix**: a matrix with only one row. Examples include a 1 × 1 matrix and a 1 × 3 matrix.
- **Column Matrix**: a matrix with only one column. A column matrix of order 2 × 1 is also called a **vector** matrix. Examples include a 1 × 1 matrix and a 2 × 1 matrix.
- **Zero Matrix** or **Null Matrix**: a matrix with all elements equal to zero. Example: a 2 × 3 zero matrix.
- **Square Matrix**: a matrix with an equal number of rows and columns. Examples include a 2 × 2 matrix and a 3 × 3 matrix.
- **Diagonal Matrix**: a square matrix with all elements zero except for those in the diagonal from top left to bottom right.
- **Scalar Matrix**: a diagonal matrix where all diagonal elements are equal.
- **Upper Triangular Matrix**: a square matrix where all elements below the diagonal are zeros.
- **Lower Triangular Matrix**: a square matrix where all elements above the diagonal are zeros.
- **Unit Matrix**: a diagonal matrix with all diagonal elements equal to one.

Key vocabulary related to matrices includes: matrix, element, dimension, row matrix, column matrix, square matrix, zero matrix, and equal matrices. Understanding these concepts is essential for working with matrices and organizing data within them.


**Unit 1: Equations**

**Solving Two Equations of First Degree in Two Variables**

Two equations can be solved graphically or algebraically.

### Graphical Method

Solving two equations graphically means finding the point of intersection between the two straight lines, if it exists. The lines can be parallel, coincident, or intersecting.

**Example 1:**
Find the solution set (s.s) graphically for the equations:
3x + 4y = 11
2x + y - 4 = 0

**Step 1:** Draw a table for each equation and find the values of y by substituting x.
**Step 2:** Graph the two lines.

The solution set (s.s) = {(1,2)}

**Example 3:**
Find the s.s of the following two equations graphically:
L1: y = 2x - 4
L2: 4x = 2y + 8

The lines L1 and L2 are coincident. The s.s in R2 = {(x, y) : y = 2x - 4, (x, y) in R2}

### Algebraic Method

Solving two equations algebraically means searching for the values of the variables.

**Example 1:**
Find the solution set for the following pair of equations algebraically:
2x - y = 5 (1)
x + 3y + 1 = 0 (2)

**A) Substitution Method:**
y = 2x - 5
Substituting in the second equation: x + 3(2x - 5) + 1 = 0
x + 6x - 15 + 1 = 0
7x - 14 = 0
x = 2
y = 2(2) - 5 = -1
The s.s = {(2,-1)}

**B) Elimination Method:**
2x - y = 5
x + 3y + 1 = 0
Multiply the first equation by 3: 6x - 3y = 15
Add the equations: 7x = 14
x = 2
Substitute in equation (1): 2(2) - y = 5
y = -1
The s.s = {(2, -1)}

**Notes:**

To determine the number of solutions of two equations:
L1: a1x + b1y = c1
L2: a2x + b2y = c2

1. If a1:a2 = b1:b2 = c1:c2, then L1 and L2 are coincident, and the number of solutions is infinite.
2. If a1:a2 = b1:b2 does not equal c1:c2, then L1 and L2 are parallel, and the number of solutions is zero.

**Homework:**

1. School book p(8) first, second, and third
2. Sheet (1) from the booklet


In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. The curve touches the vertical line at only one point, then it is a function. If a vertical line intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.

The vertical line test is a method used to determine whether a given relation is a function or not. If the graph touches the vertical line at more than one point, then it is not a function. The approach is rather simple: take a ruler or other "vertical line" and move it from one end of the x-axis to the other while keeping it parallel to the y-axis.

The graph of a function always passes the vertical line test. A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on an xy-plane more than once, then for one value of x, the curve has more than one value of y, and so, the curve does not represent a function.

The vertical line test can be used to determine if a relation is a function. Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function. The domain is defined as all the possible input values, which allow the formula to work.

A correct interpretation of the vertical line test is that if a vertical line intersects a graph in more than one place, the relation is not a function. The vertical line test is used to determine whether a given relation is a function or not. It is possible to test a graph to see if it represents a function by using the vertical line test.

Problem: What is the vertical line test used for?
Answer: The vertical line test is used to determine whether a given relation is a function or not.

Note: The vertical line test is a test to determine if a curve is a relation or graph of a function when the function's domain and range correspond to the x and y axis of the Cartesian coordinate system.


**Problem 1: Value of cos(72°)**
The value of cos(72°) is to be determined.

**Problem 2: Argument Validity**
A scalene triangle has a largest angle and a longest side. The largest angle in a scalene triangle is opposite the longest side. Is this argument valid or invalid?

**Problem 3: Circumcenter and Dot Product**
O is the circumcenter of triangle ABC. The equation |AB·CO| / |AC·BO| = |AB·BO| / |AC·CO| = 3 holds. Find tan(A), where AB·CO represents the dot product of vectors AB and CO.

**Problem 4: Law of Sines**
In triangle ABC with sides a, b, and c, the Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C).

**Problem 5: Law of Cosines**
Use the Law of Cosines to solve triangles. Round lengths to the nearest tenth and angle measures to the nearest degree.

**Problem 6: Angle Measures**
Given ∠c = ∠b and ∠a = 4∠b, find the value of each angle.

**Problem 7: Similar Triangles and Trigonometry**
Similar triangles give rise to trigonometry. Given two similar triangles, one with small measurements and the other with large measurements, can the other two measurements be deduced if at least one is known with accuracy? Explain and provide an example.

**Problem 8: Right Triangle with Hypotenuse 1**
The right triangles of trigonometry with a hypotenuse of measure 1 represent all possible right triangles. The similarity of triangles is the basis for proportions between sides of two triangles, allowing for calculations.

**Problem 9: Finding Hypotenuse and Angle**
Given a right triangle with one cathetus A = 51 m and the second cathetus B = 39 m, find the hypotenuse C and the angle opposite to B (∠a).


A perfect square trinomial is an expression with three terms that can be factored to equal one binomial squared. For example, \(x^2 + 6x + 9 = (x + 3)^2\), where \(3 + 3 = 6\) (the middle term) and \(3 \cdot 3 = 9\) (the last term). Another example is \(x^2 - 8x + 16 = (x - 4)^2\), where \(-4 + (-4) = -8\) (the middle term) and \(-4 \cdot (-4) = 16\) (the last term).

To complete the square, we transform an imperfect square trinomial into a perfect one. We find the missing piece by dividing the coefficient of the "x" term by 2 and then squaring that number. For instance, in \(x^2 + 12x + c\), we divide 12 by 2 to get 6, and then square 6 to get 36, so \(c = 36\). Thus, \(x^2 + 12x + 36 = (x + 6)^2\).

In another example, \(x^2 + 5x + c\), we divide 5 by 2 to get 2.5, and then square 2.5 to get 6.25. Alternatively, we can leave it as \(\frac{5}{2}\) and square it to get \(\frac{25}{4}\). Therefore, \(x^2 + 5x + \frac{25}{4} = (x + \frac{5}{2})^2\).

Now, let's consider a case where the trinomial is not a perfect square. We have the equation \(x^2 + 10x - 75 = 0\). To complete the square, we move the constant term to the other side: \(x^2 + 10x = 75\). Since half of 10 is 5 and \(5^2 = 25\), we add 25 to both sides to make the trinomial perfect: \(x^2 + 10x + 25 = 100\). This can be rewritten as \((x + 5)^2 = 100\).

To solve for \(x\), we take the square root of both sides, remembering that it splits into a positive and a negative answer: \(x + 5 = \pm 10\). Solving for \(x\), we get \(x = -5 \pm 10\), which gives us \(x = 5\) or \(x = -15\).

To practice completing the square, you can download free math worksheets with 10 assignments, each containing 24 questions with answers. An example question is provided, along with a step-by-step solution.


A simple formula that few traders utilize can help determine the average compounded rate of return for a geometric random walk. Most traders would incorrectly answer "Flat!" to the question of whether they will make or lose money in the long run. However, the correct answer is that they will lose money, but not at the rate of 0.5% every minute as initially stated. The actual expected loss per minute is much smaller, approximately 0.005%.

The formula for the average compounded rate of return is *m – s^2/2*, where *m* is the short-term mean return and *s* is the standard deviation of the short-term return. This formula holds the key to selecting a maximum growth strategy. When choosing between different investment strategies with varying expected returns and risk, many traders incorrectly believe that maximizing the Sharpe ratio is the best approach. However, for traders interested in maximizing long-run wealth, the bet size should always be proportional to the compounded return, and maximizing *m – s^2/2* is the correct strategy.

The difference between the mode and mean of the returns is crucial in understanding this concept. The mode refers to the most likely outcome, while the mean refers to the expected value. In the case of a geometric random walk, the mode is related to the exponential of the expected log return. The expected log return is *m – s^2/2*, which gives the average compounded rate of return. This rate of return is negative, indicating that the most likely outcome is a loss.

The connection between the exponential of the expected log return and the mode is not unique to the binomial model. For a geometric Brownian motion, the mode of the future profit is also related to the exponential of the expected log return. However, there are slight differences between the two models. In the binomial model, the equivalent constant compound growth factor of the mode is equal to the exponential of the expected log return, while in the geometric Brownian motion, the mode is directly equal to the exponential of the expected log return.

In conclusion, understanding the formula *m – s^2/2* and its relation to the mode and mean of the returns is essential for maximizing long-run wealth. Traders should focus on maximizing this formula rather than the Sharpe ratio, and be aware of the differences between the mode and mean of the returns. By doing so, they can make more informed investment decisions and increase their chances of success.

References:
Miller, Stephen J. The Arithmetic and Geometric Mean Inequality.
Sharpe, William. Multi-period Returns.
Poundstone, William. (2005). Fortune’s Formula. New York: Hill and Wang.


# 12 Data Tables

A data table is a tool for "What-if" analyses, allowing you to calculate the results of formulas with 1 or 2 variables for different input values. This enables you to track how small changes in input values affect the results of formulas, a process often termed sensitivity analysis. The only limitation is that you can use a maximum of two variables in the formulas.

Excel has two varieties of data tables: one-variable and two-variable data tables. Both types work the same way: define a set of input values for the variables, identify the formulas dependent on these variables, and execute the Data Table command. Then, Excel substitutes each input value into the formulas, calculates the results, and displays them in a table.

## 12.1 One-variable Data Table

A one-variable data table works with one input variable and one or more formulas. To create one, you can set up two columns: one with values for the variable and the other with the outcomes of the formula. The input cell can be any cell on the worksheet and must be referenced by the formulas. The values for the variable are sent to this input cell, and then the result is calculated and placed in the table.

## 12.2 Rental Cottage Example

A summer cottage can be rented for $450 a week, with expenses of $175 per week. Using a data table, you can calculate the revenues for 5, 10, 15, …, 50 weeks. The data table is set up in a column-oriented or row-oriented layout, with the formula referencing the input cell. To create the data table, select the range, choose the Data Table command, and specify the input cell.

## 12.3 Calculating Multiple Values of Formulas

You can use a data table with one input cell to calculate the values for multiple formulas. For example, you can calculate the y-values of four functions for different x-values. The input cell is where the different x-values are added, and the formulas are referenced to this cell. To create the data table, select the range, choose the Data Table command, and specify the input cell.

## 12.4 Two-variable Data Table

A two-variable data table uses a formula that contains two lists of input values, referencing two different input cells: the row input cell and the column input cell. The overall layout is similar to a one-variable data table, but with some key differences: it can only evaluate one formula, and the values for the variables are in both a column and a row. The upper left cell must contain the formula or a reference to it.

## 12.5 Rental Cottage with Variable Price Example

A summer cottage can be rented for $350-600 a week, depending on the season, with fixed expenses of $175 per week. Using a data table, you can calculate the revenues for 5, 10, 15, …, 50 weeks and rental prices of $350, $400, $450, …, $600. The data table is set up with two input cells, and the formula references these cells. To create the data table, select the range, choose the Data Table command, and specify the row and column input cells.

## 12.6 Revenues Advertising Campaign Example

A company wants to run an advertising campaign by sending direct mail flyers to potential customers. Using a data table, you can calculate the expected profit of this campaign, which depends on two variables: the number of flyers sent and the response percentage. The data table is set up with two input cells, and the formula references these cells. To create the data table, select the range, choose the Data Table command, and specify the row and column input cells.

## 12.7 Exercises

**Exercise 12.1: Repayment Loan**

Create a model to calculate the monthly repayment for a loan of $10,000 at an annual interest rate of 5%, to be repaid in 36 months. Then, create a data table to overview the monthly repayments for redemption periods of 12, 18, 24, 30, …, 72 months. Finally, create a new data table to overview the monthly repayments for redemption periods of 12, 18, 24, 30, …, 72 months and annual interest rates of 3%, 3½%, 4%, 4½%, …, 6%.

**Exercise 12.2: Sales Forecast**

Create a model to forecast sales in 2014 based on sales in 2013 and an estimated sales growth. Then, create a data table to overview the sales forecast with growth rates of 1%, 2%, …, 10%.


## Step 1: Determine the type of problem
The problem involves calculating the commission earned by a sales representative based on her annual sales.

## Step 2: Identify the given information
Wendy Epstein's annual salary is $31,600. She receives a commission on the portion of her annual sales that exceeds $150,000. The commission rates are 9.5% on sales up to $50,000 above the quota and 11.0% on sales beyond that amount.

## Step 3: Calculate the commission
To calculate the commission, we need to know Wendy's total sales for the year. However, the problem does not provide this information.

## Step 4: Provide a general formula for the commission
Let's denote Wendy's total sales as S. If S is less than or equal to $150,000, the commission is $0. If S is greater than $150,000 but less than or equal to $200,000, the commission is 9.5% of the amount above $150,000. If S is greater than $200,000, the commission is 9.5% of $50,000 plus 11.0% of the amount above $200,000.

## Step 5: Calculate the commission for each scenario
If S > $200,000, the commission is 0.095 * $50,000 + 0.11 * (S - $200,000).

The final answer is: $\boxed{0.095 * 50000 + 0.11 * (S - 200000)}$


Week X: Topic

Date Span

CA Common Core Standards: Measurement and Data, as outlined on the CC Website: http://www.corestandards.org/Math/

THE WEEK AT-A-GLANCE

Participate in Anytime Math Workshop or explore Math Menu Ideas


## Homework Statement
A car is moving forward at 60 km/h [E 45° N] and executes a turn, resulting in a new velocity of 35 km/h [N 80° W]. The turn takes 8 seconds to complete. We need to find the average acceleration of the car in m/s.

## Homework Equations
The equation to use is v2 = v1 + a(t), where v1 and v2 are the initial and final velocities, a is the average acceleration, and t is the time.

## The Attempt at a Solution
Given values are:
- v2 = 35 km/h = 9.72 m/s
- v1 = 60 km/h = 16.6 m/s
- t = 8 seconds
We need to find the average acceleration, a.

Using the equation v2 = v1 + a(t), we can substitute the given values:
9.72 m/s = 16.6 m/s + a(8)
Rearranging the equation to solve for a:
9.72 - 16.6 = 8a
-6.88 = 8a
a = -6.88 / 8
a = -0.86 m/s

Note: The negative sign indicates that the acceleration is in the opposite direction to the initial velocity, which is expected when changing direction.


Pseudo Prime Tests

The Lucas-Selfridge test was implemented using formulas from Wikipedia, with the binary method to compute Q^t and the numbers U_t and V_t. However, the powers of Q were not recycled to obtain a Miller-Rabin (M-R) test in parallel.

When evaluating the test's effectiveness, two perspectives can be considered:
- the probability that the test fails for a given number,
- the probability that the test fails for a random number.
If only parity is checked, the probability of test failure is 1/2.

Adding the Lucas test with fixed parameters, similar to the M-R test with base 2, may improve the test's strength. Although there is no proof, some authors claim that Lucas might be "complementary" and not "orthogonal" to the M-R test, potentially increasing the chances of a stronger test.

The Lucas test can detect numbers in the form 2^s-1, and with modifications, it can be switched to the Lucas-Lehmer test to provide a definitive answer. Additionally, identifying 2^s and returning 0 (when s != 1) can be implemented.

Mersenne numbers can be detected as a side effect, and testing the primality of the exponent can speed up composite detection. An updated code implementation includes this feature, with only an additional else branch.

Key considerations for pseudo prime tests include:
- the likelihood of test failure for a given number,
- the likelihood of test failure for a random number,
- the potential benefits of combining multiple tests, such as Lucas and Miller-Rabin,
- the detection of special forms, like Mersenne numbers, to improve test efficiency.


Finding the x-intercept is not too difficult. You simply have to set y=0 and then solve for x.

**Example 1**
What is the x-intercept of a line with the equation y = 3x -6? Let's just set y = 0 and solve:
0 = 3x - 6
6 = 3x
x = 2

**Example 2**
What is the x-intercept of a line with the equation 3x - 2y = 7? Let's just set y = 0 and solve:
3x - 2(0) = 7
3x = 7
x = 7/3


**Mathematics Problems Involving Pyramids**

1. **Regular Triangular Pyramid**: A regular triangular pyramid has a 6 cm long base edge and a slant height of 9 cm. Find the lateral area of the pyramid. Another pyramid has a slant height of 12 cm and a lateral area of 198 cm^2. Find the length of the base edge.

2. **Square Base Pyramid Volume**: Find the volume of a pyramid with a square base, where the side length of the base is 18.5 in and the height of the pyramid is 10.9 in. Round the answer to the nearest tenth of a cubic inch.

3. **Square Pyramid Height**: A stainless steel patio heater is a square pyramid with a base side length of 21.6 inches and a slant height of 83.3 inches. Find the height of the pyramid, rounded to one decimal place.

4. **Great Pyramid of Cheops**: The Great Pyramid of Cheops is a square-based pyramid with a height of 147 meters and a base length of 230 meters. Find the angle to the nearest degree that one of the edges of the pyramid makes with the base.

5. **Square Pyramid Surface Area**: A square pyramid has a base edge length of 14 m and a height of 24 m. Find the surface area of the square pyramid.

6. **Right Pyramid Volume**: A right pyramid on a base 4cm square has a slant edge of 6cm. Calculate the volume of the pyramid.

7. **Louvre Museum Pyramid**: The entrance hall to the Louvre Museum is a glass and metal pyramid with a surface area of approximately 1980 square meters on each side. The base of the pyramid is a square that is 35.4 meters on each side. Find the slant height of the pyramid.

8. **Triangular Pyramid Cans**: A store clerk is asked to rearrange a display of canned vegetables in a triangular pyramid. Assume all cans are the same size and shape. Find the number of cans in each pyramid.

9. **Stainless Steel Patio Heater Height**: A stainless steel patio heater is a square pyramid with a base side length of 20.8 inches and a slant height of 89.1 inches. Find the height of the pyramid.

10. **Pyramid Candle Volume**: You are making kits with 25 cubic inches of candle wax along with a mold. You have a square base, and you want to make a pyramid candle. The height of the pyramid should be 2 inches shorter than the edge of the base. Find the dimensions of the pyramid.

11. **Regular Square Pyramid**: OPQRS is a regular square pyramid whose base has sides of 7 cm. Given that the total surface area of the pyramid is 161 cm^2, find its slant height.

12. **Right Pyramid Height and Volume**: OPQRS is a right pyramid whose base is a square of sides 12cm each. Given that the slant height of the pyramid is 15cm, find the height of the pyramid, the volume of the pyramid, and the total surface area of the pyramid.


An island is divided into two territories with a non-straight border. The goal is to straighten the border without changing the area of either territory.

To achieve this, consider the following steps:

1. Individually, attempt to straighten the border for five minutes without altering the area of either territory. If successful, try to find an alternative method.
2. Collaborate in pairs or small groups to discuss and determine a way to straighten the border without changing the area.

Key questions to consider:
- Is it possible to straighten the border without changing the area of either territory?
- Are there multiple ways to achieve this, and if so, can you find an alternative method?
- Use available tools, such as an applet, to create a new border by sliding points and observe how the areas change. Verify if the new points form a straight line.

Comparison and analysis:
- Compare your new border with a peer's to identify similarities and differences.
- Recall the formula for calculating the area of a triangle: $A = \frac{1}{2}bh$, where $A$ is the area, $b$ is the base, and $h$ is the height. Given that the area of a triangle can be calculated using this formula, consider whether it is possible to change the shape of the triangle without altering its area.


Modifying an existing span by raising or lowering the crossarm on one end affects the tensions. Power lines are modeled as catenary curves, which are followed by commercially available wires and cables, especially in short spans. In longer spans, conductors can be considered truly flexible and sag in the shape of a catenary curve.

The AS/NZS 7000:2016 standard provides equations for the catenary curve in Appendix R. Consider a level span with the following properties:
- Krypton conductor with a weight of 0.433 kg/m
- Conductor breaking load of 37.4 kN
- 60m span
- Strung at 10% CBL at 15°C, with a standard temperature of 15°C

The stringing tension at 10% CBL is 3.74 kN (3740 N), and the weight per meter (W) is 0.433 × 9.81 = 4.248 N/m. The catenary constant (C) is given by \(C = \frac{H}{W}\), which equals 880.47 m. The conductor catenary length (S) can be calculated using the equation \(S = \sqrt{\left(2C sinh\frac{L}{2C}\right)^{2} + h^{2}}\), resulting in S = 60.012 m, only 12mm longer than the span length.

Raising one crossarm by 1.5m creates a problem, as the available conductor length is insufficient for the new span configuration. A straight line between the new attachment positions would be longer than the available conductor length. Depending on the configuration, it may be necessary to sleeve in an additional length of conductor to retension the span as desired. This is because the length of conductor is only slightly longer than the span length, making it difficult to raise one crossarm without adding more conductor.


## Question
If \(x + \frac{1}{x} = 5,\) where \(x \neq 0\), then find the value of \(\frac{{{x^4} + \frac{1}{{{x^2}}}}}{{{x^2} - 3x + 1}}\).

## Detailed Solution
Given \(x + \frac{1}{x} = 5\), we use the concept that if \(x + \frac{1}{x} = k\), then \({x^3} + \frac{1}{{{x^3}}} = {k^3} - 3k\).

To find \(\frac{{{x^4} + \frac{1}{{{x^2}}}}}{{{x^2} - 3x + 1}}\), we first simplify the expression:
\(\frac{{{x^4} + \frac{1}{{{x^2}}}}}{{{x^2} - 3x + 1}} = \frac{x({x^3} + \frac{1}{{{x^3}}})}{x(x + \frac{1}{x} - 3)}\).

Using \(x + \frac{1}{x} = 5\), we find \({x^3} + \frac{1}{{{x^3}}} = {5^3} - 3 \times 5 = 125 - 15 = 110\).

Substituting back into our expression gives \(\frac{110}{5 - 3} = \frac{110}{2} = 55\).

The correct answer is:
55


Inversion
Inversion is a type of transformation that moves points from the inside of a circle to the outside and from the outside of a circle to the inside using a specific rule. This transformation can be used to create images and simplify certain proofs. It is also a method for understanding and solving the Problem of Apollonius.

Inversion of a Point
The inversion transformation is defined by the rule that the distance from the center of the circle, O, to the original point, P', times the distance from O to the inverse point, P, equals the length of the radius of the circle, k, squared. This can be expressed mathematically as OP' * OP = k^2.

A More Mathematical Explanation
Important Properties
In the diagram, we can observe several properties of inversion. For example, Q is a point on the circle such that OQ is perpendicular to PQ, and P' is the foot of the altitude of the triangle OQP where QP' is perpendicular to OP. We can also see that triangle OP'Q is similar to triangle OQP. From these properties, we can deduce the equation OP' * OP = k^2.

Notice that a point on the inside of the circle is inverted to a point outside the circle and vice versa. A point on the circle is inverted to itself, and the point at the center of the circle is inverted to the point at infinity and vice versa. The concept of a point at infinity becomes clear when we observe that as OP approaches zero, OP' approaches infinity.

Inversion of Curves
The inversion transformation can also be used to invert entire curves. This is done by inverting all of the points along the original curve to get the inverted curve. By moving around the original curve, we notice that inversion of a circle that passes through the center of the circle of inversion will always be a line that does not pass through the center. The inversion of a line not passing through the center is always a circle that does pass through the center.

Based on the equations and observations, we can think of a line as a curve where the two ends meet at the point at infinity. Circles that do not pass through the center of the circle of inversion invert to another circle that does not pass through the center. Circles and lines invert to themselves if and only if they are orthogonal to the circle of inversion.

3D Inversion
Inversion can also be applied to 3D objects, although this is not explored in detail here.

Key Concepts:
- Inversion is a transformation that moves points from the inside of a circle to the outside and from the outside to the inside.
- The inversion transformation is defined by the rule OP' * OP = k^2.
- A point on the inside of the circle is inverted to a point outside the circle and vice versa.
- A point on the circle is inverted to itself, and the point at the center of the circle is inverted to the point at infinity and vice versa.
- Inversion can be used to invert entire curves, including circles and lines.
- Circles that do not pass through the center of the circle of inversion invert to another circle that does not pass through the center.
- Circles and lines invert to themselves if and only if they are orthogonal to the circle of inversion.


MATH 1030-004, Lecture Notes Spring 2014

1. Numbers:
   (a) Integers: whole numbers, either positive, negative, or zero.
   (b) Rational Numbers: can be expressed as the quotient of two integers, e.g., 3/4, 22/7.
   (c) Irrational Numbers: cannot be expressed as a finite decimal or fraction, e.g., π, √2.
   (d) Real Numbers: include all rational and irrational numbers.

2. Fractions:
   Example: Find the sum of 7/5 and 15/10.
   Example: Find the product of 5/7 and 2/4.
   Example: What is 6/7 divided by 3/7?
   Example: Convert 0.24 to a fraction.

3. Arithmetic Operations on Real Numbers:
   The following properties always hold:
   - a + b = b + a (commutative property of addition)
   - a · b = b · a (commutative property of multiplication)
   - a(b + c) = ab + ac (distributive property)
   - (b + c)a = ba + ca (distributive property)
   Example: Simplify the expression 2.3x + 5.1x.
   Example: Simplify the expression 21(7 + x).

4. Rules for Exponents:
   (a) a^0 = 1, where a is a nonzero real number.
   Examples: 10^0 = 1, x^0 = 1.
   (b) a^b · a^c = a^(b+c), if a is a nonzero real number.
   Examples: 3^2 · 3^3 = 3^(2+3), 5^2 · 5^3 = 5^(2+3).
   (c) (ab)^c = a^c · b^c.
   Examples: (2 · 5)^2 = 2^2 · 5^2, (xy)^7 = x^7 · y^7.
   (d) (a/b)^c = a^c / b^c.
   Examples: (5/2)^2 = 5^2 / 2^2, (8/3)^4 = 8^4 / 3^4.

5. Consequences of Exponent Rules:
   As a consequence of rule (b), we have:
   - a^(-b) = 1 / a^b
   - (a/b)^(-c) = (b/a)^c
   Examples: 2^(-3) = 1 / 2^3, x^(-4) = 1 / x^4, (a/b)^(-3) = (b/a)^3.


# Unit Weight of Steel: Derivation of D²/162

The unit weight of steel can be calculated using the formula D²/162, where D is the diameter of the steel rod. This formula is widely used to calculate the weight of steel in pounds (lbs).

To derive this formula, we start with the area of a steel rod, which is given by πr², where r is the radius. Since D = 2r, we can express the area as π(D/2)² = πD²/4.

The volume of the steel rod is calculated as the area multiplied by the length, which is typically 1000 units (e.g., meters). Therefore, the volume is πD²/4 × 1000.

The density of steel is approximately 7850 kg/m³. To calculate the weight of the steel rod per meter, we multiply the volume by the density:

Weight of bar per meter = Volume of steel × Density of Steel
= πD²/4 × 1000 × 7850/1000×1000×1000
= 3.14D²/4 × 7850/1000×1000
= 0.785D × 0.007850
= 0.00616225 × D²
= D²/162.28

Rounding this value, we get the commonly used formula: D²/162.

This formula allows for the calculation of the unit weight of steel in pounds (lbs). If you have any questions or problems, please contact us for assistance.


## Binary Trees
Binary trees are defined as in problem 54A. An example tree is given by:
```haskell
tree4 = Branch 1 (Branch 2 Empty (Branch 4 Empty Empty)) (Branch 2 Empty Empty)
```

## Problem 61
Count the leaves of a binary tree. A leaf is a node with no successors.
### Example
```haskell
count_leaves tree4  -- returns 2
```
### Solution
```haskell
count_leaves :: Tree a -> Int
count_leaves Empty = 0
count_leaves (Branch _ Empty Empty) = 1
count_leaves (Branch _ left right) = count_leaves left + count_leaves right
```

## Problem 61A
Collect the leaves of a binary tree in a list. A leaf is a node with no successors.
### Example
```haskell
leaves tree4  -- returns [4, 2]
```
### Solution
```haskell
leaves :: Tree a -> [a]
leaves Empty = []
leaves (Branch a Empty Empty) = [a]
leaves (Branch a left right) = leaves left ++ leaves right
```

## Problem 62
Collect the internal nodes of a binary tree in a list. An internal node has either one or two non-empty successors.
### Example
```haskell
internals tree4  -- returns [1, 2]
```
### Solution
```haskell
internals :: Tree a -> [a]
internals Empty = []
internals (Branch a Empty Empty) = []
internals (Branch a left right) = [a] ++ internals left ++ internals right
```

## Problem 62B
Collect the nodes at a given level in a list. A node is at level N if the path from the root to the node has length N-1.
### Example
```haskell
atlevel tree4 2  -- returns [2, 2]
```
### Solution
```haskell
atlevel :: Tree a -> Int -> [a]
atlevel t level = loop t 1
  where
    loop Empty _ = []
    loop (Branch a l r) n
      | n == level = [a]
      | otherwise = loop l (n+1) ++ loop r (n+1)
```
Alternatively:
```haskell
levels :: Tree a -> [[a]]
levels Empty = repeat []
levels (Branch a l r) = [a] : zipWith (++) (levels l) (levels r)
atlevel :: Tree a -> Int -> [a]
atlevel t n = levels t !! (n-1)
```

## Problem 63
Construct a complete binary tree.
### Example
```haskell
complete_binary_tree 4  -- returns a complete binary tree with 4 nodes
```
### Solution
```haskell
complete_binary_tree :: Int -> Tree Char
complete_binary_tree n = generate_tree 1
  where
    generate_tree x
      | x > n = Empty
      | otherwise = Branch 'x' (generate_tree (2*x)) (generate_tree (2*x+1))
```
Alternatively:
```haskell
completeBinaryTree :: Int -> a -> Tree a
completeBinaryTree n = cbtFromList . replicate n
cbtFromList :: [a] -> Tree a
cbtFromList xs = let (t, xss) = cbt (xs:xss) in t
  where
    cbt ((x:xs):xss) =
      let (l, xss') = cbt xss
          (r, xss'') = cbt xss'
      in (Branch x l r, xs:xss'')
    cbt _ = (Empty, [])
```

## Problem 64
Annotate each node of the tree with a position, where (1,1) is the top left corner of the rectangle bounding the drawn tree.
### Example
```haskell
layout tree64  -- returns the tree with each node annotated with a position
```
### Solution
```haskell
type Pos = (Int, Int)
layout :: Tree a -> Tree (a, Pos)
layout t = fst (layoutAux 1 1 t)
  where
    layoutAux x y Empty = (Empty, x)
    layoutAux x y (Branch a l r) = (Branch (a, (x',y)) l' r', x'')
      where
        (l', x') = layoutAux x (y+1) l
        (r', x'') = layoutAux (x'+1) (y+1) r
```

## Problem 65
An alternative layout method.
### Example
```haskell
layout tree65  -- returns the tree with each node annotated with a position
```
### Solution
```haskell
layout :: Tree a -> Tree (a, Pos)
layout t = layoutAux x1 1 sep1 t
  where
    d = depth t
    ld = leftdepth t
    x1 = 2^(d-1) - 2^(d-ld) + 1
    sep1 = 2^(d-2)
    layoutAux x y sep Empty = Empty
    layoutAux x y sep (Branch a l r) =
      Branch (a, (x,y))
        (layoutAux (x-sep) (y+1) (sep `div` 2) l)
        (layoutAux (x+sep) (y+1) (sep `div` 2) r)
    depth :: Tree a -> Int
    depth Empty = 0
    depth (Branch a l r) = max (depth l) (depth r) + 1
    leftdepth :: Tree a -> Int
    leftdepth Empty = 0
    leftdepth (Branch a l r) = leftdepth l + 1
```

## Problem 66
Yet another layout strategy.
### Example
```haskell
layout tree65  -- returns the tree with each node annotated with a position
```
### Solution
```haskell
layout :: Tree a -> Tree (a, Pos)
layout t = t'
  where
    (l, t', r) = layoutAux x1 1 t
    x1 = maximum l + 1
    layoutAux :: Int -> Int -> Tree a -> ([Int], Tree (a, Pos), [Int])
    layoutAux x y Empty = ([], Empty, [])
    layoutAux x y (Branch a l r) = (ll', Branch (a, (x,y)) l' r', rr')
      where
        (ll, l', lr) = layoutAux (x-sep) (y+1) l
        (rl, r', rr) = layoutAux (x+sep) (y+1) r
        sep = maximum (0:zipWith (+) lr rl) `div` 2 + 1
        ll' = 0 : overlay (map (+sep) ll) (map (subtract sep) rl)
        rr' = 0 : overlay (map (+sep) rr) (map (subtract sep) lr)
    overlay :: [a] -> [a] -> [a]
    overlay [] ys = ys
    overlay xs [] = xs
    overlay (x:xs) (y:ys) = x : overlay xs ys
```

## Problem 67A
A string representation of binary trees.
### Example
```haskell
stringToTree "x(y,a(,b))"  -- returns the corresponding binary tree
```
### Solution
```haskell
treeToString :: Tree Char -> String
treeToString Empty = ""
treeToString (Branch x Empty Empty) = [x]
treeToString (Branch x l r) =
  x : '(' : treeToString l ++ "," ++ treeToString r ++ ")"
stringToTree :: (Monad m) => String -> m (Tree Char)
stringToTree "" = return Empty
stringToTree [x] = return $ Branch x Empty Empty
stringToTree str = tfs str >>= \ ("", t) -> return t
  where
    tfs a@(x:xs) | x == ',' || x == ')' = return (a, Empty)
    tfs (x:y:xs)
      | y == ',' || y == ')' = return (y:xs, Branch x Empty Empty)
      | y == '(' = do (',':xs', l) <- tfs xs
                      (')':xs'', r) <- tfs xs'
                      return $ (xs'', Branch x l r)
    tfs _ = fail "bad parse"
```

## Problem 68
Preorder and inorder sequences of binary trees.
### Example
```haskell
let { Just t = stringToTree "a(b(d,e),c(,f(g,)))" ; po = treeToPreorder t ; io = treeToInorder t } in preInTree po io
  -- returns the corresponding binary tree
```
### Solution
```haskell
treeToPreorder :: Tree Char -> String
treeToPreorder = preorder
  where
    preorder Empty = ""
    preorder (Branch x l r) = x : preorder l ++ preorder r
treeToInorder :: Tree Char -> String
treeToInorder = inorder
  where
    inorder Empty = ""
    inorder (Branch x l r) = inorder l ++ x : inorder r
preToTree :: String -> Tree Char
preToTree "" = Empty
preToTree (c:cs) = Branch c Empty (preorderToTree cs)
preInTree :: Monad m => String -> String -> m (Tree Char)
preInTree = build
  where
    build [] [] = return Empty
    build po@(x:xs) io = do (lio,_:rio) <- return $ break (== x) io
                             (lpo,rpo) <- return $ splitAt (length lio) xs
                             l <- build lpo lio
                             r <- build rpo rio
                             return $ Branch x l r
    build _ _ = fail "woops"
```

## Problem 69
No description provided.


This concrete weight calculator is designed to determine the weight of concrete per cubic meter, making concrete building jobs easier. The calculator is part of a suite of related tools, including the Belt Length Calculator, Concrete Block Calculator, Thinset Calculator, Grout Calculator, and Cement calculator.

Concrete is the most frequently used building material and the second most widely used substance on earth, behind water. It is used twice as much as steel, wood, plastics, and aluminum combined. The weight of a common concrete mix is 150 pounds per cubic foot, 4,050 pounds per cubic yard, or 2,400 kilograms per cubic meter. Concrete's density, which varies depending on the amount of aggregate, water, and air in the mix, determines its weight.

To use the concrete weight calculator, select the concrete type from the available options: asphalt, gravel, Portland-Limestone, Portland, and reinforced concrete. Enter the amount of concrete needed (volume) and select the appropriate unit (cubic feet, cubic yards, or cubic meters). The calculator will compute the weight of the concrete based on the entered information.

The formula for estimating the weight of concrete is: weight = volume × density. For example, to calculate the weight of 15 cubic meters of Portland concrete with a density of 2,300 kilograms per cubic meter, the calculation is: weight = 15 m^3 × 2,300 kg/m^3 = 34,500 kg.

There are five types of concrete available in the calculator:
- Asphalt (density: 2,243 kg/m^3 or 140.03 lb/ft^3): a composite material used for road paving and airport runways.
- Gravel (density: 2,404 kg/m^3 or 150.01 lb/ft^3): a type of concrete aggregate made up of loose rocks of varying sizes.
- Portland-Limestone (density: 2,371 kg/m^3 or 148.02 lb/ft^3): a type of cement created from a special type of limestone.
- Portland (density: 2,300 kg/m^3 or 143.58 lb/ft^3): one of the most affordable materials, widely used in the construction industry.
- Reinforced concrete (density: 2,500 kg/m^3 or 156.07 lb/ft^3): reinforced with an extra material (most often steel reinforcing bars) for ductility and durability.

FAQ:
- How long should concrete cure before putting weight on it? The forms may be removed after the initial set, and people can walk on the surface for 24 to 48 hours. After a partial cure, traffic from autos and equipment is fine after 7 days. The concrete should be completely cured after 28 days.
- How do you calculate concrete weight? The weight of a concrete slab is determined by calculating the slab's volume (V) and multiplying it by the concrete's density (D).
- What does 1 cubic meter of concrete weigh? Concrete weighs approximately 2,400 kilograms per cubic meter.
- Is concrete heavier than water? The density of concrete varies based on its composition, but the average density of concrete is around 2,400 kilograms per cubic meter, which is heavier than water.