# Quintic Function

A quintic function is a polynomial of degree five, defined by the equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$, where $a$, $b$, $c$, $d$, $e$, and $f$ are members of a field, typically the rational numbers, real numbers, or complex numbers, and $a$ is nonzero.

## Properties of Quintic Functions

Quintic functions have an odd degree, which means they appear similar to cubic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a quintic function is a quartic function.

## Quintic Equations

A quintic equation is an equation of the form $ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$. Solving quintic equations in terms of radicals was a major problem in algebra from the 16th century until the 19th century, when the impossibility of such a general solution was proved with the Abel-Ruffini theorem.

## Finding Roots of a Quintic Equation

Finding the roots of a given polynomial has been a prominent mathematical problem. Solving linear, quadratic, cubic, and quartic equations by factorization into radicals can always be done, but there is no algebraic expression for general quintic equations over the rationals in terms of radicals.

## Solvable Quintics

Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is reducible. A solvable quintic is an irreducible quintic polynomial whose roots may be expressed in terms of radicals.

## Characterizing Solvable Quintics

To characterize solvable quintics, Évariste Galois developed techniques that gave rise to group theory and Galois theory. Applying these techniques, Arthur Cayley found a general criterion for determining whether any given quintic is solvable.

## Roots of a Solvable Quintic

A polynomial equation is solvable by radicals if its Galois group is a solvable group. In the case of irreducible quintics, the Galois group is a subgroup of the symmetric group $S_5$ of all permutations of a five-element set, which is solvable if and only if it is a subgroup of the group $F_5$ of order 20.

## Beyond Radicals

About 1835, Jerrard demonstrated that quintics can be solved by using ultraradicals, also known as Bring radicals. In 1858, Charles Hermite showed that the Bring radical could be characterized in terms of the Jacobi theta functions and their associated elliptic modular functions.

## Application to Celestial Mechanics

Solving for the locations of the Lagrangian points of an astronomical orbit in which the masses of both objects are non-negligible involves solving a quintic. The locations of $L_2$ and $L_1$ are the solutions to the following equations, where the gravitational forces of two masses on a third provide the satellite's centripetal force necessary to be in a synchronous orbit with Earth around the Sun.

## See Also

* Galois theory
* Group theory
* Elliptic modular functions
* Jacobi theta functions
* Bring radical
* Ultraradicals

## Notes

* Michele Elia and Piero Filipponi. "Equations of the Bring-Jerrard form, the golden section, and square Fibonacci numbers", *Fibonacci Quarterly* 36, June-July 1998, 282–286.
* A. Cayley. "On a new auxiliary equation in the theory of equation of the fifth order", *Philosophical Transactions of the Royal Society of London* (1861).
* George Paxton Young. "Solvable Quintics Equations with Commensurable Coefficients", *American Journal of Mathematics* 10 (1888), 99–130.
* David S. Dummit. *Solving Solvable Quintics*.
* Felix Klein. *Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree*, trans. George Gavin Morrice, Trübner & Co., 1888.

## References

* Charles Hermite, "Sur la résolution de l'équation du cinquème degré", *Œuvres de Charles Hermite*, t.2, pp. 5–21, Gauthier-Villars, 1908.
* Felix Klein, *Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree*, trans. George Gavin Morrice, Trübner & Co., 1888.
* Leopold Kronecker, "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite", *Comptes Rendus de l'Académie des Sciences*, t. XLVI, 1858 (1), pp. 1150–1152.
* Blair Spearman and Kenneth S. Williams, "Characterization of solvable quintics $x^5+ax+b$", *American Mathematical Monthly*, Vol. 101 (1994), pp. 986–992.
* Ian Stewart, *Galois Theory* 2nd Edition, Chapman and Hall, 1989.
* Jörg Bewersdorff, *Galois theory for beginners: A historical perspective*, American Mathematical Society, 2006.

## External Links

* Mathworld - Quintic Equation – more details on methods for solving Quintics.
* Solving Solvable Quintics – a method for solving solvable quintics due to David S. Dummit.
* A method for removing all intermediate terms from a given equation - a recent English translation of Tschirnhaus' 1683 paper.


The harmonic mean is calculated using the `harmmean` function, denoted as `m = harmmean(X)`. This function calculates the harmonic mean of a sample, where `X` can be a vector, matrix, or N-dimensional array.

For vectors, `harmmean(X)` returns the harmonic mean of the elements in `X`. For matrices, `harmmean(X)` returns a row vector containing the harmonic means of each column. For N-dimensional arrays, `harmmean` operates along the first nonsingleton dimension of `X`.

The `harmmean` function can also be used to calculate the harmonic mean of all elements in `X` by specifying the `'all'` option, denoted as `m = harmmean(X,'all')`. Additionally, the harmonic mean can be calculated along a specific dimension `dim` of `X` using `m = harmmean(X,dim)`.

The function also allows for the specification of multiple dimensions using a vector `vecdim`, denoted as `m = harmmean(X,vecdim)`. Each element of `vecdim` represents a dimension of the input array `X`. The output `m` has a length of 1 in the specified operating dimensions, while the other dimension lengths remain the same as `X`.

For example, if `X` is a 2-by-3-by-4 array, then `harmmean(X,[1 2])` returns a 1-by-1-by-4 array, where each element is the harmonic mean of the elements on the corresponding page of `X`.

The `harmmean` function also includes an option to exclude `NaN` values from the calculation, denoted as `m = harmmean(___,nanflag)`. By default, `harmmean` includes `NaN` values in the calculation, but setting `nanflag` to `'omitnan'` excludes them. When `harmmean` computes the harmonic mean of an array containing `0`, the returned value is `0`.


The four roots of 1 can be represented as i, -1, -i, and 1, corresponding to 1^(1/4), 1^(2/4), 1^(3/4), and 1^(4/4) respectively. When adding the dimensional exponents of 1 in each case, representing the degree of rotation with respect to the circular circumference, a pattern emerges. Values from 90 to 270 degrees are treated as negative, while values in the 1st and 4th quadrants are positive.

If n, representing the fractional values associated with the n roots of 1, is divisible by 4, the sum of corresponding values equals 0. For example, with the four roots, the sum of fractions is 1/4 - 2/4 - 3/4 + 4/4 = 0.

For the sum of even numbered fractions where n is not divisible by 4, the corresponding sum equals 1/2. For instance, when n = 2, the sum of fractions is -1/2 + 1 = 1/2.

When n is one less than a number divisible by 4, the total sum equals 0. For example, with n = 3, the sum of fractions is -1/3 - 2/3 + 3/3 = 0.

When n is one more than a number divisible by 4, the total sum equals 1. For example, when n = 5, the sum of fractions is 1/5 - 2/5 - 3/5 + 4/5 + 5/5 = 1.


To illustrate the concept, let's consider a small ball thrown straight up, with its position measured perfectly every tenth of a second. We can create a graph with time on the horizontal axis and the ball's height on the vertical axis. The ball starts at 4 feet off the ground and reaches a height of 20 feet. The graph shows a positive slope until it reaches the highest point, and then the slope becomes negative. Finding the highest point involves finding when the derivative, or slope function, equals zero.

The equation for the ball's height is h(t) = -16t^2 + 32t + 4, where height is in feet and time is in seconds. The -16 is determined by the force of gravity on earth, and the 32 is determined by the initial upward velocity. To find the slope of the tangent line at a given point, we need to find the change in y and the change in x, which requires two points. However, we only have one point, so we're stuck.

One strategy is to use a second point on the graph very near the first and create a secant line that goes through both points. By getting closer and closer, we can approximate the tangent line. For example, if we use t=0.4 and t=0.5, we can find the slope of the secant line. To find the exact slope, we need to imagine the second point getting infinitely close to the first, which is where calculus comes in.

This concept is based on the idea of taking the limit as the distance between the time values, h, approaches zero. The algebra can get complex, but it's easier to understand for a generic time t. We can imagine t represents one time, and t+h represents another time very close to it. The height at time t+h would be h(t+h) = -16(t+h)^2 + 32(t+h) + 4.

Finding the slope involves finding the average rate of change between the two points and then taking the limit as h approaches zero. This concept is crucial in calculus, as it allows us to find instantaneous speeds and understand how functions change over time. The equation for the slope is:

m = (h(t+h) - h(t)) / h

As h approaches zero, the slope approaches the instantaneous rate of change. This concept is powerful and has numerous applications in physics, engineering, and other fields.

Calculus is a branch of mathematics that deals with the study of continuous change, and it's based on the concept of limits. The use of infinity is what separates calculus from algebra, and it allows us to model real-world phenomena with greater accuracy. While the algebra can get complex, the underlying concepts are intuitive and powerful, making calculus a beautiful and useful subject to study.


## Area of a Square or a Rectangle

The area of a plane figure is a measure of the space inside the figure and is two-dimensional. It is measured in square units, such as square inches, square feet, square centimeters, or square meters.

### Formulas

- For a square with side *s*, the area is given by *s*^{2}.
- For a rectangle with length *l* and width *w*, the area is *l* × *w*.

### Examples

1. **Finding the Area of a Square**:
   - Given: The side of the square is 5 m.
   - Solution:
     - Step 1: Identify the side of the square, which is 5 m.
     - Step 2: Calculate the area using the formula *s*^{2} = (5m)^{2} = 25 m^{2}.

2. **Finding the Area of a Rectangle**:
   - Given: The length of the rectangle is 8 cm, and the width is 4 cm.
   - Solution:
     - Step 1: Identify the length and width of the rectangle, which are 8 cm and 4 cm, respectively.
     - Step 2: Calculate the area using the formula *l* × *w* = 8 × 4 = 32 square cm.


A Bode diagram is a graphical representation of a complex-valued function's amplitude gain and phase shift as a function of frequency. It is named after Hendrik Wade Bode, who used these diagrams in his work at Bell Laboratories in the 1930s. Bode diagrams are used to represent linear time-invariant systems (LTI) in electronics, electrical engineering, control engineering, and mechatronics, as well as in impedance spectroscopy.

The Bode diagram describes the relationship between a harmonic excitation at the input of a system and the associated output signal in the steady state. To fully describe an LTI system, multiple diagrams are needed. The Bode diagram is used to represent the transmission behavior of a dynamic system, also known as frequency response or frequency response. Other diagram forms, such as the Nyquist diagram or the pole-zero diagram, serve other purposes, including stability considerations.

The Bode diagram has several characteristic properties:
- The x-axis represents frequency or angular frequency, logarithmically scaled, allowing for the behavior of the system to be seen over a large frequency range at a glance.
- The y-axis of the first graph represents the amplitude gain, in decibels or logarithmic scaling, and is called the amplitude response.
- The y-axis of the second graph represents the phase shift, linearly scaled, and is called the phase response.
- The amplitude and phase response are plotted on top of each other, so that the gain and phase of a frequency are vertically above each other.

Due to the logarithmic scaling of the amplitude response, Bode diagrams have the advantage that complex diagrams can be created from the overlaying of simple partial diagrams. This corresponds to a series connection of transmission elements. The complex function is broken down into sub-functions of the first and second order by factoring, and the gain is plotted logarithmically, allowing for the multiplication of partial functions to become the addition of their amplitude responses.

The following table summarizes the transfer function, designation, amplitude response, phase response, and Bode diagram for various elements:
| Transfer function | Designation | Amplitude response | Phase response | Bode diagram |
| --- | --- | --- | --- | --- |
| P element |  |  |  |  |
| D link | +20 dB/decade, 0 dB at  | constant at  |  |  |
| I-link | -20 dB/decade, 0 dB at  | constant at  |  |  |
| PD link | Kink at , then +20 dB/decade | from 0 to  over two decades, at  |  |  |
| PT1 element | Kink at , then -20 dB/decade | from 0 to  over two decades, at  |  |  |
| PT2 element | Kink at , then -40 dB/decade | from 0 to  over two decades with a compression depending on d |  |  |

Using the example of a PT1 system, the advantages of a logarithmic representation can be illustrated. A simple low-pass, such as an RC element, forms a PT1 system. The corner frequency or limit frequency can be read from the linearly divided diagram, but with more complex systems, it makes more sense to work in the double logarithmic Bode diagram.

In the Bode diagram, the course of the function can also be idealized with straight lines. The corner frequency is at the intersection of the horizontal and descending straight lines. The real function has already dropped by -3 dB at this point. If the system shows proportional behavior, the gain can be read off the y-axis.

A system can be identified based on the gradient and phase progression. With a PT1 system, the gradient above is -1:1. A doubling of the frequency leads to a halving (-6 dB) of the amplitude, and a tenfold increase in frequency reduces the gain to a tenth, i.e., -20 dB. The phase at  is -45° and for  it is -90°.

If two PT1 systems are connected in series, the result is a PT2 system with damping. Above the first corner frequency, the slope is -1:1, and after the second corner frequency, the slope is -2:1. If the two corner frequencies are far enough apart, the phase is -45° at the corner frequency and -90° at the second.

A PT2S system capable of oscillation can be represented with a complex pole or as a second-order polynomial. Above the corner frequency, the slope is -2:1. The phase is -90° at the corner frequency and tends towards -180° at infinity. A resonance increase occurs depending on the damping.

With integrators, called I-systems, there is no horizontal straight line segment for small frequencies. It starts immediately with a slope of -1:1. Correspondingly, with a differentiator, called a D system, the slope is immediately +1:1. The integration or differentiation time constant can be read off for. This can also be viewed as reinforcement, as systems basically only have P, I, or D behavior.


## Homework Statement
An electromagnet operates at 50A with a coil inductance of 100H. Determine the voltage developed when the switch is opened and closed, given the switch operates in 1 millisecond.

## Homework Equations
The current equation is I(t) = Vb/(R(1-e^(-tR/L))).

## The Attempt at a Solution
To proceed, we need to address the missing resistance value in the equation I(t) = Vb/(R(1-e^(-tR/L))). Without resistance, we cannot directly calculate the voltage using the provided formula. We must either find the resistance or use an alternative approach that does not require knowing the resistance, such as considering the relationship between voltage, inductance, and current change over time when the switch is opened or closed.


Problem H: Cent Savings

When making a purchase, the supermarket rounds the total price to the nearest multiple of 10 cents, with 5 cents being rounded upwards. For example, 94 cents are rounded to 90 cents, while 95 cents are rounded to 100 cents.

The goal is to divide the purchase into groups and pay for each part separately to minimize the total cost. There are $d$ dividers available to divide the purchase into up to $d+1$ groups. The items cannot be rearranged on the conveyor belt.

Input:
- One line with two integers $n$ ($1 \leq n \le 2,000$) and $d$ ($1 \leq d \le 20$), representing the number of items and the number of available dividers.
- One line with $n$ integers $p_1, \ldots, p_n$ ($1 \leq p_i \leq 10,000$ for $1 \leq i \leq n$), representing the prices of the items in cents, in the order they appear on the belt.

Output:
The minimum amount of money needed to buy all the items, using up to $d$ dividers.

Sample Input 1:
5 1
13 21 55 60 42
Sample Output 1:
190

Sample Input 2:
5 2
1 1 1 1 1
Sample Output 2:
0


# 3 Ways to Understand Matrix Multiplication

Matrix multiplication can be understood from three points of view: columns, rows, and combinations of both. This post describes these interpretations, discussing the definition of matrix multiplication, and providing examples and facts to help test intuition.

The usual definition of matrix multiplication is as a summation or dot product of rows of A and columns of B. The dot product of row 1 of A and column 1 of B gives the first entry of C. In general, the ij-th entry of C is the i-th row of A dotted with the j-th column of B.

**Interpretation 1: Columns**
Each column of C is a linear combination of the columns of A, with weights given by columns of B. The columns of C are linear combinations of the columns of A, with weights given by columns of B. This means:
- A matrix multiplied by a vector is a linear combination of the columns of the matrix.
- The columns of C are the results of applying a linear transformation to columns of B.
- If the columns of B are linearly independent, then C has a column of zeros if and only if A has a column of zeros.
- The span of the columns of C is contained in the span of the columns of A, so rank(AB) ≤ rank(A).

**Interpretation 2: Rows**
The rows of C are linear combinations of the rows of B, with weights given by rows of A. This means:
- If the rows of AB are linearly independent, then so are the rows of B.
- If A has a row of zeros, then AB has a row of zeros.
- The span of the rows of AB contains the span of the rows of B.
- If A is invertible, then AB has the same row space as B.

**Interpretation 3: Columns and Rows**
The matrix C is the sum of matrices that are columns of A multiplied by rows of B. Each matrix in the sum has columns that are scalar multiples of a column of A. This means:
- Each matrix in the sum has a 1-dimensional column space.
- Interchanging two columns of A and the corresponding rows of B results in the same product AB.

The dimension of the span of the columns of a matrix is called its column rank, and the dimension of the span of the rows is called its row rank. The column rank and row rank of a matrix are equal. If A and B are square matrices and AB = I, then BA = I, so B is the inverse of A.

In conclusion, understanding matrix multiplication from different points of view provides insights into its properties and behavior. These interpretations can be used to review basic facts about matrix multiplication, independence, and span.


**Problem 1:**
The amount of pepper, p, in milligrams, used in seasoning is modeled by the equation p = 2s + 1, where s is the amount of salt in milligrams. According to the model, for every 1 milligram increase in salt, the amount of pepper added is determined by the coefficient of s, which is 2. This means 2 milligrams of pepper are added for every 1 milligram increase in salt.

**Problem 2:**
A survey of doctors and lawyers shows the number of vacations taken each year. The probability of a doctor taking at least 1 vacation is 4/5. To find the number of people taking at least 2 vacations, we need the total number of doctors and the number of doctors taking 0 vacations. Assuming there are 5 doctors and 4 take at least 1 vacation, 1 takes 0 vacations. The table shows 3 doctors take 2 vacations and 1 takes 3 vacations, so 4 people (3 doctors and 1 lawyer) take at least 2 vacations.

**Problem 3:**
The cost per necklace, C, is determined by the equation C = 10 + 5n, where n is the number of necklaces made.
(a) The cost of making 6 necklaces is C = 10 + 5*6 = 10 + 30 = 40 dollars.
(b) To make a total profit of $564, the total revenue must be the total cost plus the profit. If each necklace is sold for $150, the total revenue from selling n necklaces is 150n. The total cost is 10 + 5n. So, 150n = (10 + 5n) + 564. Simplifying, 150n = 5n + 574, 145n = 574, n = 574 / 145 = 4.

**Problem 4:**
The inequality is not provided, so we cannot determine which value of x satisfies it.

**Problem 5:**
No problem or solution is provided.


Infinite products are mathematical expressions that involve the multiplication of an infinite number of terms. These products have various applications in mathematics, particularly in number theory and algebra.

One example of an infinite product is the Seidel product, which is given by:

(2)

This product is a special case of the more general product:

(3)

Another example is the Viète product, which is given by:

(4)

This product can be derived from the infinite series:

(5)

The product:

(6)

is based on the fact that every positive integer has a unique representation in binary.

Euler products are another type of infinite product, which are based on the expansion of the Riemann zeta function. One example is:

(7)

Setting s=2 gives:

(8)

Setting s=4 gives:

The left side is equal to:

One way of seeing this is by taking the term in the infinite product expansion. This gives:

Hence:

Finally, one obtains:

(9)

by dividing:

by:

(8)

The product:

(11)

can be proved using unique factorisation of the odd natural numbers.

The product:

(13)

can be obtained by dividing 3/4 times:

(8)

by the square of:

(11)

The product:

(14)

can be obtained by dividing:

by:

(13)

The product:

(15)

involves the constant e and can be derived using the infinite sum form of e:

The product:

(16)

can be easily obtained from:

(15)

by multiplying both sides by:

The pentagonal number theorem is a result due to Euler, which shows how much cancellation there is when the product is expanded. The coefficient of:

is the number of partitions of n into an even number of parts minus the number of partitions of n into an odd number of parts.

Proofs of these products can be found using various methods, including geometric series and Euler products. The references provided give more information on the proofs and derivations of these products.

The key concepts involved in infinite products include:

* Geometric series
* Euler products
* Riemann zeta function
* Unique factorisation
* Binary representation
* Infinite series
* Constant e
* Pentagonal numbers

Important numbers and formulas include:

* Seidel product:
(2)
* Viète product:
(4)
* Euler product:
(7)
* Riemann zeta function:
* Constant e:
* Pentagonal number theorem:

These infinite products have various applications in mathematics and are an interesting area of study.


#### Answer

The statement makes sense.

#### Work Step by Step

In an augmented matrix, any row with all zero elements can be deleted.
Consider the augmented matrix:
$\left[ \begin{matrix}
1 & -1 & -2 & 2 \\
0 & 1 & -10 & -1 \\
0 & 0 & 0 & 0 \\
\end{matrix} \right]$
The last row represents the equation $0x+0y+0z=0$, which is always true since $0=0$. This implies that variable z can take any value, resulting in infinitely many solutions for the system of linear equations. Therefore, row 3 can be deleted from the matrix.


A **rational function** is a function that can be represented as the quotient of two polynomials. The general forms of rational functions are:

$$ f(x) = \frac{ax + b}{cx^2 + dx + e} $$ and $$ f(x) = \frac{ax^2 + bx + c}{dx + e} $$

The first form represents a rational function where the numerator is a linear polynomial and the denominator is a quadratic polynomial. The second form represents a rational function where both the numerator and the denominator are polynomials of degree 2 and 1 respectively.

Examples of rational functions include:

$$ f(x) = \frac{2x + 3}{x^2 - 5x + 6} $$ and $$ f(x) = \frac{x^2 + 4x + 4}{3x + 6} $$

The domain of a rational function is all real numbers except where the denominator equals zero. **Vertical Asymptotes** occur where the function becomes undefined, found by setting the denominator to zero and solving for \(x\).

**Horizontal Asymptotes** describe the behavior of the graph as \(x\) approaches positive or negative infinity. To determine the horizontal asymptote, compare the degrees of the numerator and denominator:

- If the degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is the horizontal asymptote.
- If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator by one, there is no horizontal asymptote, and you should check for an **oblique asymptote**.

An **oblique (or Slant) Asymptote** occurs when the degree of the numerator is exactly one more than the degree of the denominator. To find the equation of the oblique asymptote, perform polynomial long division. The quotient (ignoring the remainder) gives the equation of the oblique asymptote.

Algebraic division, or polynomial long division, is a method for dividing a polynomial by another polynomial of the same or lower degree. The key formula is:

$$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$

This technique is useful in simplifying rational expressions and solving polynomial equations.


# GEOMETRY EXAMINATION

## INSTRUCTIONS

This test consists of 50 multiple-choice questions. Choose the best answer from the five options labeled A, B, C, D, and E. The test will be scored as follows: 5 points for each correct answer, 1 point for each unanswered question, and 0 points for each incorrect answer.

## KEY INFORMATION

* All variables and constants represent real numbers, except those indicated otherwise.
* Diagrams are not necessarily to scale.
* Geometric notation:
	+ |AB| is the segment between points A and B.
	+ |AB| is the distance between points A and B.
	+ m∠A is the measure of angle A in degrees.
	+ is the arc between points A and B on a circle.

## PROBLEMS

1. In a right square pyramid, all five faces have the same area. If the height of the pyramid is 9 inches, what is the total surface area in square inches?
2. Two regular polygons have the same number of sides, with side lengths 48 meters and 55 meters. What is the length of the side of another regular polygon with the same number of sides, if its area is equal to the sum of the areas of the first two polygons?
3. A 160-foot rope is suspended between two 100-foot poles. If the lowest point of the rope is 72 feet above the ground, what is the distance between the poles?
4. In a circle with center O and radius 1, a chord AB has a length equal to the radius. A perpendicular from O to AB meets AB at M, and a perpendicular from M to OA meets OA at D. What is the area of triangle AMD?
5. In a circle, AB = 24 and the perpendicular chord is 4 times as long as AB. How long is the perpendicular chord?
6. If the diameter of a cylindrical can is increased by 30%, by what percentage should the height be increased to triple the volume of the can?
7. Find the circumference of a circle inscribed in an isosceles right triangle with legs of length 1.
8. In a figure, ∠ADE = 100° and DE = DC = CB = BA. What is the measure of ∠EAD?
9. What is the length of the hypotenuse of a right triangle with an area of 40 and a longest leg of length b?
10. In a rectangular prism, the areas of the faces are 18, 40, and 80 square units. What is the length of the diagonal?
11. Find the possible values of k so that the lines kx + y = 3 and x - y = 2 intersect in the first quadrant.
12. A circle is inscribed in a triangle with sides of length 4 cm, 13 cm, and 15 cm. Find the area of the circle.
13. How many faces does a polyhedral solid with 36 vertices and 64 edges have?
14. A cylindrical tank has a spiral staircase attached to its exterior. The staircase makes exactly 4 complete revolutions and is 1 foot wide. If the tank is 20 feet high and has a diameter of 16 feet, what is the length of the exterior edge of the staircase?
15. Find the radius of a circle if the difference between the areas of the inscribed and circumscribed equilateral triangles is 25 square inches.
16. Find the area of a trapezoid with bases of 17 meters and 42 meters and legs of 15 meters and 20 meters.
17. The sides of a triangle are the roots of a quadratic equation, and they are all natural numbers. What is the area of the triangle?
18. Three circles of equal radius touch each other and enclose a three-cornered concave area. What is the area of this region?
19. A cylinder is cut on a slant, with a height of 15 inches to 21 inches and a radius of 4 inches. Find the volume of the cylinder.
20. If ABCD is a square with E on DC such that DE = 2 and EC = 4, what is the perimeter of triangle AEB?
21. A semicircle has a diameter AB, and a rectangle CDEF is inscribed in the semicircle with CD = 24 and DE = 56. A square FGHI with side x is between the rectangle and the semicircle. What is the area of FGHI?
22. The length, width, and height of a rectangular solid are in the ratio of 6:8:24. If the diagonal of the solid is inches long, what is the length of the longest side of the solid?
23. In a figure, a rectangle has a width of 6 and a height of 5. Circle A has radius r, and circle B has radius 1. Find the value of r.
24. The center of a circle defined by an equation is (h, k). What is the product of h and k?
25. AB and CD are perpendicular to a diameter of a circle O. Let CM be a chord that intersects AB at E, so that CE = 6 and EM = 5. Find the circumference of the circle.
26. Given a square ABCD with M as the midpoint of DC, what is the ratio of the area of triangle MEC to the area of quadrilateral AEMD?
27. Determine the area outside the square |x| + |y| ≤ 1 and inside the circle x^2 + y^2 = 1.
28. AB is a diameter of a circle, and CD is a chord parallel to AB. Find the angle ∠C - ∠D.
29. The side of a square is a. Find the length of a side of a regular octagon obtained from the square by cutting off its corners.
30. In a figure, two diameters of a circle intersect at the center C. A chord intersects the diameters at point G. If ∠BCE has a measure of 60°, what is the measure of ∠AGF?
31. A line is tangent to a circle centered at the origin. If the line intersects the circle at a point, what is the length of the line segment from the origin to the point of tangency?
32. Sam sees the top of a 15-foot lamp, which is 250 feet away, lining up perfectly with the peak of a distant mountain. If the mountain is 15 miles away, what is the best estimate of the height of the mountain relative to Sam?
33. In a figure, how many different paths are there from S to E if one never visits the same point twice?
34. In a figure, six similar triangles share one side with the next triangle and all share one vertex. All angles at that vertex measure 60°. If the side of the last triangle is 1/6 as large as the longest side of the first triangle, how many times larger is the area of the largest triangle compared to the smallest?
35. Find the height of a square pyramid formed by four equilateral triangles with sides of length 2.
36. In triangle ABC, the measure of angle A is twice the measure of angle B, and the measure of angle B is three times the measure of angle C. What is the measure of angle B?
37. In a figure, the area of rectangle PQST is 9 square inches. If the degree measure of ∠PRQ is 60°, what is the length of side PQ?
38. Let A be the point (3, 2), and B be the reflection of A about the x-axis. Let C be the reflection point of B about the line y = -x, and D be the reflection point of C about the origin. What is the area of quadrilateral ABCD?
39. A and B are two points on a circle with radius r and center O. If the distance between A and B is r, what is the radian measure of ∠AOB?
40. In a figure, PQR is a 3-4-5 right triangle with right angle at P. Points X, Y, and Z on PQ, QR, and PR are chosen so that PXYZ is a square. What is the length of segment PX?
41. Given a right triangle with right angle at point C and legs of length a and b, what is the length of the segment that joins C to the hypotenuse and bisects the angle at C?
42. Equilateral triangle ABC has vertices A, B, and C, and center D. A new triangle A'B'C is formed where A' is the midpoint of AD, B' is the midpoint of BD, and C' is the midpoint of CD. What is the ratio of the area of triangle ABC to triangle A'B'C?
43. Let C be the circle described by (a - x)^2 + y^2 = r^2, where 0 < r < a. Let m be the slope of the line through the origin that is tangent to C at a point in the first quadrant. What is m?
44. In a triangle, the medians to the sides are perpendicular. Find the area of the triangle.
45. In a three-dimensional rectangular coordinate system, find the total surface area of the solid defined by |x| + |y| + |z| ≤ 1.
46. If the graphs of 2y + x + 3 = 0 and 3y + ax + 2 = 0 are to meet at a right angle, what is the value of a?
47. A pair of opposite vertices and the midpoints of a pair of opposite edges of a cube are connected to form a quadrilateral. If each edge of the cube has length k, what is the area of the quadrilateral?
48. You have 6 sticks of lengths 10, 20, 30, 40, 50, and 60 centimeters. How many non-congruent triangles can be formed by choosing three of the sticks to make the sides?
49. A glass box of 7 cm × 12 cm × 18 cm is partly filled with colored water. When the box is placed on one of its 7 × 12 sides, the water level is 15 cm above the table. If the box is placed on one of its 7 × 18 sides, what is the water level above the table?
50. In a figure, ABEDC is circumscribed by a circle through points A, D, and E. ABC is an equilateral triangle with side length 2, and BCDE is a square. Find the radius of the circle.


**Supplies: FCAT Reference Sheet • Plain Graph Paper**

## 6th Grade Math

The 6th grade math curriculum is divided into four quarters, each covering various topics in mathematics.

### Quarter 1: Expressions, Fractions, and Formulas
- Writing and evaluating expressions
- Properties of addition and multiplication
- Distributive property
- Using formulas to solve problems
- Estimating with fractions
- Multiplying and dividing fractions and mixed numbers
- Fraction-decimal equivalency

### Quarter 2: Decimals, Percents, and Data Analysis
- Multiplying and dividing decimals
- Comparing and ordering fractions, decimals, and percents
- Finding the percent of a number
- Estimating with percentages
- Ratios and rates
- Mean, median, mode, and range
- Describing, analyzing, and summarizing a data set

### Quarter 3: Geometry, Equations, and Problem-Solving
- Concept of pi and circumference of a circle
- Pi and area of circles
- Perimeter and area of composite figures
- Writing equations in one variable
- Solving equations using addition, subtraction, multiplication, and division
- Solving two-step equations
- Finding missing dimensions in plane figures and prisms

### Quarter 4: Functions, Graphs, and Inequalities
- Introduction to functions
- Input-output tables
- Analyzing graphs
- Test-taking tips
- Solving inequalities using addition, subtraction, multiplication, and division
- Understanding the number line and the coordinate plane


# What are 10 clues as to why the Earth is not flat?

**Earth Curvature**

The Earth's shape corresponds to a sphere, deviating from a tangential plane over short distances. This curvature affects altitude measurements, which must be corrected accordingly. The fact that the Earth is approximately spherical was known to Ionian scientists around 600 BC.

**Calculation of Earth's Curvature**

With an average Earth radius of 6371 km, the ideal Earth surface deviates from a tangential plane as follows:

* 0.8 mm at 100 m
* 20 mm at 500 m
* 78 mm at 1000 m
* 1.96 m at 5000 m
* 7.85 m at 10,000 m

A simple approximation formula is y = -L^2 / (2R), where L is the distance, R is the Earth radius, and y is the deviation in meters. For example, two people 10,000 m apart would need to be at an eye level of 1.96 m to have visual contact.

**Graphic Illustration**

A graphic illustrates the curvature of the Earth, with two points 1000 km apart and at an altitude of 19.6 km. A more accurate approximation formula is y = -√(L^2 + R^2) - R, which gives the following values:

* 1.96 m at 5 km
* 7.84 m at 10 km
* 196 m at 50 km
* 784 m at 100 km
* 1,764 m at 150 km
* 3,135 m at 200 km
* 4,898 m at 250 km

**Correction of Altitude Measurements**

The correction of altitude measurements due to the Earth's curvature is essential, even on short distances, and grows quadratically with distance. When surveying locations, the curvature of the Earth only has an effect at greater distances.

**Practical Example**

The elevation angle determination of mountains, such as Mont Blanc (4810 m), results in the following angles of altitude depending on the distance:

* at 50 km: 5.27° (5.49° without curvature)
* at 100 km: 2.30° (2.75° without curvature)
* at 150 km: 1.16° (1.83° without curvature)
* at 200 km: 0.48° (1.38° without curvature)
* at 250 km: (0.02°)

The 250 km value indicates that at this distance, the tip of Mont Blanc is below the "horizon line". Terrestrial refraction also plays a role, breaking light rays in the direction of the Earth's curvature, and can be interpreted as reducing the effect of the curvature by 5-15%, depending on meteorological conditions.


Power is defined as the time rate of doing work or expending energy, calculated as P = W/t, where P is power, W is work, and t is time. This can also be expressed in units of watts (W) or joules per second (J/s).

The equation P = W/t can be related to voltage (V) and charge (q) using the formula V = W/q. By cross-multiplying, we get W = Vq. Substituting this into the power equation gives P = Vq/t. Since q/t represents current (I), we can simplify this to P = VI.

Using Ohm's law, V = IR, we can substitute to get P = I^2R. Alternatively, rearranging Ohm's law to I = V/R and substituting into the power equation gives P = V^2/R. Therefore, the three equivalent equations for power are P = VI = I^2R = V^2/R.

Given the equation P = W/t, we can calculate work as W = VIt. For example, if a circuit has a voltage of 12V, a current of 0.60A, and a time of 60 seconds, the work done is W = 12V * 0.60A * 60s = 432 J, which is close to the given answer of 430 J.

In a series circuit with a potential difference of 1.5V and a current of 0.10A, the rate of energy use can be calculated using the equation P = VI. Substituting the given values, we get P = 1.5V * 0.10A = 0.15W. This represents the rate at which the circuit uses energy.


# Maths practice for 14 year olds

#### Number of problems found: 225

- Today 2

A desk is being sold at a 72% discount, with a sale price of $100.80. What was the original price?

If you pay $11 for 1 apple and 9 bananas and $7 for 3 apples and 1 banana, how much does an apple and a banana cost?

Rebecca is knitting a scarf that is 1 1/2 feet long, which is ⅔ of the desired length. How long will the scarf be when finished? Write a multiplication and a division equation to represent this situation.

A fruit company uses two fertilizers, brand A and brand B. Brand A contains 8 pounds of nitrogen and 4 pounds of phosphoric acid, while brand B contains 7 pounds of nitrogen and 6 pounds of phosphoric acid.

Mr. Akpalu buys 3 kg of sugar, 10 kg of wheat, and 1 kg of salt. The prices are €20, €10, and €6 per kg at one shop, and €15 and €8 per kg at another.

If 1 1/2 accountants can do 2 1/2 tax returns in 1 1/2 hours, how many tax returns can 3 accountants do in 3 hours?

Two teachers are paid 3,000 pesos in the ratio 3:5. How much does each receive?

Madam Lowa buys 4 cakes and 4 puffs for $16. Each cake costs $0.50 more than each puff. How much does 1 puff cost?

Rina's cell phone bills for the past five months are $91, $69, $74, $56, and $70. What is her average cell phone bill?

There are 72 tickets to be shared between Section X and Section Y in the ratio 4:5. How many tickets will Section Y receive?

Eight painters paint 15,200 m² in 5 days. How many painters will paint 13,680 m² in 3 days?

A cyclist rides uphill at 20 km/h and downhill at 60 km/h. The distance from the summit to the destination is 30 km. What is the average speed?

Calculate the volume of a sphere with a radius of 5 cm and the area of the spherical canopy with a radius of 4 cm.

The temperature rises 14°C in 3 1/2 hours. What is the rate of temperature rise per hour?

Four people can sit in four free seats in how many different ways if Rebecca wants to sit with John?

A jacket has a 20% discount, with an original price of Php 1,250. What is the change if Php 1,000 is paid?

The price of a notebook is 20% less than the price of a book. By what percent is the price of the book higher than the notebook?

A washing machine is on sale for R 1,240, representing a 22.5% discount. What was the original price?

An amusement park sells child and adult tickets at a ratio of 8:1. If 147 more child tickets are sold than adult tickets, how many tickets are sold in total?

The average of two numbers is 9.5, and one number is 11. What is the value of the other number?

Do you have a math question or word problem that you can't solve? Ask a question or post a math problem to get help.


## Calculating Square Roots

To calculate the square root of a number, start by making an initial guess. Divide the number by your guess, resulting in two numbers that multiply to get the original number. Take the average of these two numbers to obtain a second guess for the square root.

## Manual Calculation of Square Roots

For manual calculation, group the digits of the number in pairs starting from the right end. For example, to calculate the square root of 8254129, write it as 8 25 41 29 and proceed with long division.

## Perfect Squares

A perfect square is the product of a number multiplied by itself, denoted as n × n. Examples include:

| Integer | Perfect Square |
| --- | --- |
| 2 x 2 | 4 |
| 3 x 3 | 9 |
| 4 x 4 | 16 |
| 5 x 5 | 25 |

## Shortcut for Square Root Symbol

The Alt code for the square root symbol is Alt + 251 or 221A, followed by Alt + X. To insert the symbol, position the pointer, press and hold the Alt key, type 251, and then release.

## Shortcut to Find Squares

There are shortcut methods to find squares, involving the following steps:

| Column I | Column II | Column III |
| --- | --- | --- |
| a^2 | 2 x a x b | b^2 |
| 25 + 7 | 70 + 4 = 74 | 49 |

This method simplifies the process of finding squares by breaking it down into manageable columns.


A tutorial movie and website on Topology and Geometry are available at http://www.dimensions-math.org/Dim_regarder_E.htm. It is recommended to watch the movie before reading the text on the web pages, which can be found at http://www.dimensions-math.org/Dim_tour_E.htm (tour/guide) and http://www.dimensions-math.org/Dim_chap_E.htm (details). The creators, Jos Leys, Étienne Ghys, and Aurélien Alvarez, are to be congratulated.

The movie is divided into nine chapters:
- **Chapter 1**: Dimension two, an elementary introduction suitable for secondary school students, exploring the concept of the Earth as a ball.
- **Chapter 2**: Dimension three, still elementary but requiring imagination and including philosophical elements, with exercises to check understanding.
- **Chapters 3 and 4**: Introduction to the fourth dimension, more challenging and potentially complex, with recommendations to pause and review as needed.
- **Chapters 5 and 6**: Complex numbers, providing an introduction or refresher, with chapter 6 ending in a deep zoom scene.
- **Chapters 7 and 8**: The Hopf fibration, a more advanced topic not typically taught in secondary school or early university years, explained in the film with additional references for support.
- **Chapter 9**: A special chapter proving a theorem of geometry using only secondary school level mathematics, emphasizing the importance of proofs in mathematics.

Each chapter is designed to be engaging, with some offering exercises or additional resources for deeper understanding. The film and website combination provides a comprehensive and visually appealing introduction to topology and geometry.


## Selina Concise Mathematics Class 10 ICSE Solutions Chapter 1 Value Added Tax Ex 1C

**Question 1.**
Madan purchases a compact computer system for ₹ 47,700, which includes 10% rebate on the marked price and then 6% Sales Tax on the remaining price. Find the marked price of the computer.
Solution:
Sales price of the computer = ₹ 47700
Rate of Sales Tax = 6%
Hence marked price = ₹ 50,000

**Question 2.**
An article is marked at ₹ 500. The wholesaler sells it to a retailer at 20% discount and charges sales-tax on the remaining price at 12.5%. The retailer, in turn, sells the article to a customer at its marked price and charges sales-tax at the same rate. Calculate:
(i) the price paid by the customer.
(ii) the amount of VAT paid by the retailer.
Solution:
Marked price of an article = ₹ 500
Rate of discount = 20%
(i) Price paid by the customer = ₹ 500 + ₹ 62.50 = ₹ 562.50
(ii) VAT paid by the retailer = ₹ 62.50 – ₹ 50.00 = ₹ 12.50

**Question 3.**
An article is marked at ₹ 4,500 and the rate of sales-tax on it is 6%. A trader buys this article at some discount and sells it to a customer at the marked price. If the trader pays ₹ 81 as VAT; find:
(i) how much per cent discount does the trader get?
(ii) the total money paid by the trader, including tax, to buy the article.
Solution:
Marked price = ₹ 4500
Rate of S.T. = 6%
VAT paid by the trader = ₹ 81
C.P. for the customer = ₹ 4500

**Question 4.**
A retailer sells an article for ₹ 5,350 including 7% Sales Tax on the listed price. If he had bought it at a discount and has made a profit of 25% on the whole, And the rate of discount he gets.
Solution:
S.P. of an article of the retailer (including S.T.) = ₹ 5350
Rate of S.T. = 7%

**Question 5.**
A shopkeeper buys a camera at a discount of 20% from the wholesaler, the printed price of the camera being ₹ 1600 and the rate of sales tax is 6%. The shopkeeper sells it to the buyer at the printed price and charges tax at the same rate. Find:
(i) The price at which the camera can be bought from the shopkeeper.
(ii) The VAT (Value Added Tax) paid by the shopkeeper.
Solution:
Printed price (M.P) = ₹ 1600
Rate of discount = 20%
(i) The buyers bought the camera = ₹ 1600 + ₹ 96 = ₹ 1696
(ii) VAT paid by the shopkeeper = ₹ 96 – ₹ 76.80 = ₹ 19.20

**Question 6.**
Tarun bought an article for ₹ 8000 and spent ₹ 1000 for transportation. He marked the article at ₹ 11,700 and sold it to a customer. If the customer had to pay 10% sales tax, find:
(i) The customer’s price
(ii) Tarun’s profit percent.
Solution:
Tarun’s cost price = ₹ 8000
Transportation charges = ₹ 1000
Tarun’s total cost = (₹ 8000 + ₹ 1000) = ₹ 9000
Tarun’s selling price = ₹ 11,700
Rate of sales tax = 10%

**Question 7.**
A shopkeeper sells an article at the listed price of ₹ 1,500 and the rate of VAT is 12% at each stage of sale. If the shopkeeper pays a VAT of ₹ 36 to the Government, what was the price, inclusive of Tax, at which the shopkeeper purchased the articles from the wholesaler?
Solution:
Tax charged by shopkeeper

**Question 8.**
A shopkeeper bought a washing machine at a discount of 20% from a wholesaler, the printed price of the washing machine being ₹ 18,000. The shopkeeper sells it to a consumer at a discount of 10% on the printed price. If the rate of sales tax under is 8%, find:
(i) the VAT paid by the shopkeeper.
(ii) the total amount that the consumer pays for the washing machine.
Solution:
(i) S.P. of washing machine
VAT paid by shopkeeper = Tax charged – Tax paid = ₹ 1296 – ₹ 1152 = ₹144
(ii) Price paid by customer = ₹ 16200 + ₹ 1296 = ₹ 17496

**Question 9.**
Mohit, a dealer in electronic goods, buys a high class TV set for ₹ 61,200. He sells this TV set to Geeta, Geeta to Rohan and Rohan sells it to Manoj. If the profit at each stage is ₹ 2,000 and the rate of VAT at each stage is 12.5%, find:
(i) total amount of tax (under VAT) paid to the Government.
(ii) Money paid by Manoj to buy the TV set.
Solution:
For Mohit,
C.P. of electronic goods = ₹ 61200
Amount of profit in each case = ₹ 2000
Net VAT = ₹ 8400 – ₹ 8150 = ₹ 250
(i) Total amount of VAT paid to govt. = ₹ 7650 + ₹ 250 + ₹ 250 + ₹ 250 = ₹ 8400
(ii) Money paid by Manoj to buy the T.V. set = ₹ 67200 + ₹ 8400 = ₹ 75600

**Question 10.**
A shopkeeper buys an article at a discount of 30% of the list price which is ₹ 48,000. In turn, the shopkeeper sells the article at 10% discount. If the rate of VAT is 10%, find the VAT to be paid by the shopkeeper.
Solution:
Market value (M.P.) of an article = ₹ 48000
Rate of discount = 30%
Amount of discount = ₹ 48000 x 30/100 = ₹ 14400
(i) Cost price of shopkeeper = ₹ 48000 – ₹ 14400 = ₹ 33600
(ii) C.P. for shopkeeper = ₹ 33600
Rate of VAT = 10%
Net VAT paid by the shopkeeper = VAT recovered from customer – VAT paid to dealer
= ₹ 4320 – ₹ 3360 = ₹ 960

**Question 11.**
A company sells an article to a dealer for ₹ 40,500 including VAT (sales-tax). The dealer sells it to some other dealer for ₹ 42,500 plus tax. The second dealer sells it to a customer at a profit of ₹ 3,000. If the rate of sales-tax under VAT is 8%, find:
(i) the cost of the article (excluding tax) to the first dealer.
(ii) the total tax (under VAT) received by the Government.
(iii) the amount that a customer pays for the article.
Solution:
For a company,
S.P. of an article including VAT = ₹ 40500
Rate of VAT = 8%
(i) Amount of article excluding VAT
= ₹ 425 x 108 = ₹ 45900
Total tax (VAT) = ₹ 45900 – ₹ 42500 = ₹ 3000
S.P. for the second dealer = ₹ 42500 + ₹ 3000 = ₹ 45500
(ii) VAT = ₹ (45500 x 8)/100 = ₹ 3640
(iii) Total amount paid by customer = ₹ 45500 + ₹ 3640 = ₹ 49140

**Question 12.**
A wholesaler buys a TV from the manufacturer for ₹ 25,000. He marks the price of the TV 20% above his cost price and sells it to a retailer at 10% discount on the marked price. If the rate of VAT is 8%, find the:
(i) marked price.
(ii) retailer’s cost price inclusive of tax.
(iii) VAT paid by the wholesaler.
Solution:
S.P. for the manufactures = ₹ 25000
or C.P. for a whole seller a T.V. = ₹ 25000
C.P. for the retailer = ₹ 27000 + ₹ 2160 = ₹ 29160
(iii) VAT paid by the whole seller = ₹ 2160 – ₹ 2000 = ₹ 160

**Question 13.**
A dealer buys an article at a discount of 30 % from the wholesaler, the marked price being ₹ 6,000. The dealer sells it to a shopkeeper at a discount of 10% on the marked price. If the rate of VAT is 6%, find:
(i) the price paid by the shopkeeper including the tax.
(ii) the VAT paid by the dealer.
Solution:
Market price of the article = ₹ 6000
A dealer buys an article at a discount of 30% from the wholesaler.
Price of the article which the dealer paid to the wholesaler = 6000 – 30% of 6000
Amount of the article inclusive of sales tax at which the dealer bought it
= ₹ 4200 + ₹ 252 = ₹ 4452
Dealer sells the article at a discount of 10% to the shopkeeper.
Price of the article which the shopkeeper paid to the dealer = ₹ 6000 – 10% of 6000
Amount of the article inclusive of sales tax at which the shopkeeper bought it
= ₹ 5400 + ₹ 324 = ₹ 5724
The value added by dealer = ₹ 5400 – ₹ 4200 = ₹ 1200
Amount of VAT paid by dealer = 6% of 1200
= (6/100) x 1200 = ₹ 72
Price paid by shopkeeper including tax is ₹ 5724
VAT paid by dealer is ₹ 72