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7. 11. 3 227 " 3 18 " 8. 6 1.21 " 12. 1 2 28 " 16. 48b3, b 0 14. 15. 9 27x3y5, x. 0, y. 0 $ In 17–20, in each case, solve for the variable, using the set of real numbers as the replacement set. 17. y2 81 20. 2k2 144 0 18. m2 0.09 19. 3x2 600 0.25a8b10 " 498 Operations With Radicals 21. a. Use a calculator to evaluate ... |
. What is the product of 3.5x2 and 6.2x3? A 2x Exploration STEP 1. On a sheet of graph paper, draw the positive ray of the real number line. Draw a square, using the interval from 0 to 1 as one side of the square. Draw the diagonal of this square from 0 to its opposite vertex. Cumulative Review 499 2. Why? Place the po... |
(1) 40 (2) 20 4. The product (3a 2)(2a 3) can be written as (3) 16 (4) 10 " " 5 2 5 (4) 3 " (1) 6a2 6 (2) 6a2 a 6 (3) 6a2 5a 6 (4) 6a2 5a 6 500 Operations With Radicals 5. If 0.2x 8 x 4, then x equals (1) 120 (2) 12 (3) 15 (4) 15 6. If the height of a right circular cylinder is 12 centimeters and the measure of the di... |
, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. ABCD is a trapezoid with BC'AB and BC'CD and CD = 8. A line segment is drawn from A to E, the midpoint of, AB 13, BC 12,.CD Cumulative Review 501 a. Find the area of AED. b. Fin... |
height to which the ball rises and the distance between the batter and the point where the ball hits the ground. In this chapter we will study the quadratic equation that models the path of a baseball as well as functions and relations that are not linear. Solving Quadratic Equations 503 13-1 SOLVING QUADRATIC EQUATIO... |
2 1 3 2 0 5? 5? 0 0 0 ✔ Since both 2 and 1 satisfy the equation x2 3x 2 0, the solution set of this equation is {2, 1}. The roots of the equation, that is, the values of the variable that make the equation true, are 2 and 1. Note that the factors of the trinomial x2 3x 2 are (x 2) and (x 1). If the trinomial x2 3x 2 i... |
0 0 ✔ Answer: The solution set is 0, 3 2. V U 3 Check for x : 2 2x2 3x 2 5? 3 3 2 B 9 5 Note: We never divide both sides of an equation by an expression containing a variable. If we had divided 2x2 = 3x by x, we would have obtained the equation 2x 3, whose solution is but would have lost the solution x 0. 3 2 506 Quad... |
Solving Quadratic Equations 507 Solution How to Proceed (1) Write an equation from the given information: (2) Set the equation in standard form: (3) Factor the quadratic expression: (4) Solve for r: (5) Reject the zero value. Use the positive value to write the answer. Answer The radius of the circle is 6 units. pr2 3... |
h of a ball thrown into the air with an initial vertical velocity of 48 feet per sec- ond from a height of 5 feet above the ground is given by the equation h 16t2 48t 5 where t is the time in seconds that the ball has been in the air. After how many seconds is the ball at a height of 37 feet? 46. A batter hit a baseba... |
Graph of a Quadratic Function 509 dentally, the actual path of the ball––is a curve called a parabola. The special properties of parabolas are discussed in this section. An equation of the form y ax2 bx c (a 0) is called a second-degree polynomial function or a quadratic function. It is a function because for every or... |
function y x2 4x 1, the equation of the vertical line of symor x 2. Every point on the parabola to the left of x 2 metry is matches a point on the parabola to the right of x 2, and vice versa. 2(24) 2(1) x 5 This example illustrates the following properties of the graph of the qua- dratic equation y x2 4x 1: 1. The gr... |
22) 2(21) dratic equation y 5 2x2 2 2x 1 5 : 1. The graph of the equation is a parabola. 2. The parabola is symmetric with respect to the vertical line x 1. 3. The parabola opens downward and has a maximum point at (1, 6) 4. The equation defines a function. 5. The constant term, 5, is the y-intercept. When the equation... |
5 2(0) 2(1) 0 or x 0. b. (1) Since the vertex of the parabola is on the axis of symmetry, the x-coordinate of the vertex is 0. Use three values of x that are less than 0 and three values of x that are greater than 0. Make a table using integral values of x from 3 to 3. (2) Plot the points associated with each ordered ... |
3. e. The vertex is (0, 3). 514 Quadratic Relations and Functions When the coordinates of the turning point are rational numbers, we can use from the or maximum minimum CALC menu to find the vertex. In Example 1, since the turning point is a minimum, use minimum: CALC CALC 2nd 2nd 3 4 R Q Q R ENTER: 2nd CALC 3 When th... |
the x-value of the vertex. ENTER: 2nd TBLSET 1 ENTER.5 ENTER Before creating the table, make sure that “Indpnt:” and “Depend:” are set to “auto.” If. they are not, press ENTER ENTER 2nd Finally, press to create the table. Scroll up and down to view the values of x and y. TABLE X Y1 4 1.75 0 -1.25 -2 -2.25 -2 1 1.5 2 2... |
of a Quadratic Function 517 Translating, Reflecting, and Scaling Graphs of Quadratic Functions Just as linear and absolute value functions can be translated, reflected, or scaled, graphs of quadratic functions can also be manipulated by working with the graph of the quadratic function y x2. y 5 x2 1 2.5 For instance, ... |
5 (x 1 1.5)2 2 5 Answer b. First, stretch vertically by a factor of 4: y 5 4x2 Then, translate the resulting function 2 units down: 1 c. First, compress vertically by a factor of : 6 Then, reflect in the x-axis: d. First, reflect in the x-axis: Then, translate the resulting function 2 units up: Finally, translate the ... |
paper or on a calculator, showing at least 15. y x2 6x 1 18. y x2 4x 3 21. Write an equation for the resulting function if the graph of y x2 is: 16. y x2 2x 8 19. y x2 3x 7 17. y x2 8x 12 20. y x2 x 5 a. reflected in the x-axis and shifted 3 units left. b. compressed vertically by a factor of and shifted 9 units up. 2... |
for perimeter to express the measure of a second side of the rectangle. b. Write an equation for the area of the rectangle in terms of x. c. Draw the graph of the equation written in b. d. What are the dimensions of the rectangle with maximum area? e. What is the maximum area of the rectangle? f. List four other possi... |
one number, 3, that makes the equation true. The function y x2 6x 9 has infinitely many pairs of numbers that make the equation true. The graph of this function shown at the right intersects the x-axis in only one point, (3, 0). Since the y-coordinate of this point is 0, the x-coordinate of this point is the root of t... |
}. 1 –1 1 Note that in Example 1 the quadratic expression x2 5x 6 can be factored into (x 2)(x 3), from which we can obtain the solution set 2, 3. 6 5 524 Quadratic Relations and Functions EXAMPLE 2 Use the graph of y x2 3x 4 to find the linear factors of x2 1 3x 2 4. y Solution The graph intersects the x-axis at (4, 0... |
solution set for the equation h(x) 0? Graphic Solution of a Quadratic-Linear System 525 In 13–15, refer to the graph of the parabola shown below. 13. Which of the following is the equation of the parabola? (1) y (x 2)(x 3) (2) y (x 2)(x 3) (3) y (x 2)(x 3) (4) y (x 2)(x 3) y 1 14. Set the equation of the parabola equa... |
0 6 1 6 6 4 12 6 9 18 6 16 24 6 25 30 6 36 36 2) On the same set of axes, draw the graph of y x 4. Make a table of val- ues and plot the points2, –2) y = x – 4 (5, 1) x Graphic Solution of a Quadratic-Linear System 527 The line could also have been graphed by using the y-intercept, 4. Starting at the point (0, 4) move... |
y x2 2x 3 y 0 4. y x2 2x 3 y 3 5. y x2 2x 3 y 4 6. y x2 2x 1 –1 x 7. For what values of c do the equations y x2 2x 3 and y c have no points in common? 8. a. Draw the graph of y x2 4x 2, in the interval 1 x 5. b. On the same set of axes, draw the graph of y x 2. c. Write the coordinates of the points of intersection of... |
Y2 and Y3. c. Using a calculator method, determine, to the nearest tenth of a second, when the baseball has a height of 46 feet and when the baseball hits the ground (reaches a height of 0 feet). 13-5 ALGEBRAIC SOLUTION OF A QUADRATIC-LINEAR SYSTEM = x2 – 4 x In the last section, we learned to solve a quadratic-linear... |
2x 1 1 1 2(4x, y) (4, 3) (8) Check each ordered pair in each of the given equations: Check for x2 4x 3 5 1 4 5? 2 A A 5 4 5 2x 1 1 5 1 4 5? 1 2 2 B A 5 4 5 Algebraic Solution of a Quadratic-Linear System 531 Check for x 4, y 3 y x2 4x 3 3 5? (4)2 2 4(4) 1 3 3 5? 16 2 16 1 3 ✔ 3 5 3 y 3 5? 1 1 2x 1 1 2(4) 1 1 3 5? 2 1 ... |
real numbers. 2. Explain why the equations x2 y2 49 and x 8 have no common solution in the set of real numbers. 3. y x2 2x Developing Skills In 3–20, solve each system of equations algebraically and check all solutions. 4. y x2 5 y x 5 7. x2 y 9 y x 9 y x 1 8. x2 y 2 y x 6. y x2 2x 1 y x 5. y x2 4x 3 y x 3 9. x2 2y 5 ... |
? Explain your answer. 23. The length of the diagonal of a rectangle is 85 meters. The length of the rectangle is 1 meter longer than the width. Find the dimensions of the rectangle. 24. The length of one leg of an isosceles triangle is 29 feet. The length of the altitude to the base of the triangle is 1 foot more than... |
graph of y x2 is translated by k units in the vertical direction, the equation of the image is y x2 k. When the graph of y x2 is translated k units in the horizontal direction, the equation of the image is y (x k)2. When the graph of y x2 is reflected over the x-axis, the equation of the image is y x2. The graph of y ... |
4x 1 10. f(x) x2 6x 1 In 11–16, solve each system of equations graphically and check the solution(s) if they exist. 11. y x2 6 x y 6 14. y x2 4x x y 4 y x 16. y 2x x2 x y 2 y x 5 15. y 5 x2 y 4 12. y x2 2x 1 13. y x2 x 3 In 17–22, solve each system of equations algebraically and check the solutions. 18. y x2 4x 9 17. ... |
x2 75 can be expressed as 2 8 $ (2) (3) (4) 0.4 " (1) (3x 15)(x 5) (2) (x 5)(3x 15) (3) 3x(x 5)(x 5) (4) 3(x 5)(x 5) 4. When b2 4b is subtracted from 3b2 3b the difference is (1) 3 7b (2) 2b2 7b 5. The solution set of the equation 0.5x 4 2x 0.5 is (3) 2b2 7b (4) 2b2 b (1) {3} (2) {1.4} (3) {30} (4) {14} 6. Which of the... |
base. The perimeter of the triangle is 54 centimeters, what is the measure of each side of the triangle? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all ques... |
. Items are sold at a third off, or at a fraction of the original price. B Fractions are also used when sharing. For example, Andrea designed some beautiful Ukrainian eggs this year. She gave onefifth of the eggs to her grandparents.Then she gave one-fourth of the eggs she had left to her parents. Next, she presented h... |
when the denominator, x 9, is equal to 0. x 9 0 x 9 Answer EXERCISES Writing About Mathematics 1. Since any number divided by itself equals 1, the solution set for 1 is the set of all real x x numbers. Do you agree with this statement? Explain why or why not. 2. Aaron multiplied by (equal to 1) to obtain the fraction ... |
4 a 5 1 2 5 10 and 1 2 are equivalent fractions, and both are equivalent to a 2a, when a 0. The examples shown above illustrate the division property of a fraction: if the numerator and the denominator of a fraction are divided by the same nonzero number, the resulting fraction is equal to the original fraction. In ge... |
denominator the greatest common factor determined in step 2. 4. Use the multiplication property of a fraction. METHOD 2 1. Factor both the numerator and the denominator. 2. Divide both the numerator and the denominator by their greatest common factor. Reducing Fractions to Lowest Terms 543 EXAMPLE 1 Reduce 15x2 35x4 t... |
which the fractions are not defined. 3. 4x 12x 7. ab cb 11. 15x2 5x 15. 112a2b 28ac 19. 3x 1 6 4 23. 2ax 1 2bx 6x2 27. 18b2 1 30b 9b3 31. 2a2 6a2 2 2ab 35. x2 2 1 5x 2 5 39. 16 2 a2 2a 2 8 43. x2 1 7x 1 12 x2 2 16 47. x2 2 25 x2 2 2x 2 15 51. r2 2 4r 2 5 r2 2 2r 2 15 4. 8. 12. 16. 20. 24. 27y2 36y 3ay2 6by2 5x2 25x4 2... |
2 2 50 x2 1 8x 1 15 54. x2 2 7xy 1 12y2 x2 1 xy 2 20y2 Multiplying Fractions 545 55. a. Use substitution to find the numerical value of x2 2 5x x 2 5, then reduce each numerical frac- tion to lowest terms when: (1) x 7 (4) x 2 (2) x 10 (5) x 4 (3) x 20 (6) x 10 b. What pattern, if any, do you observe for the answers to... |
terms, we may use 5x2 7y by 14y2 15x3 either of the two methods. In this example, x 0 and y 0. METHOD 1. 5x2 7y? 14y2 15x3 5 5x2? 14y2 7y? 15x3 5 70x2y2 105x3y 5 2y 3x? 35x2y 35x2y 5 2y 3x? 1 5 2y 3x METHOD 2. 5x2 7x? 14y2 15x3 5 5x2 1 7y 1? 2y 14y2 15x3 3x 5 2y 3x (the cancellation method) While Method 1 is longer, i... |
8a Solution Think of 12a as 12a 1. 12a? 8a 5 36a 8a 5 12a 3 3 1? 8a 9 9 2 5 9 5 4a 4a? 2 5 1? 2 Answer 9 2 (a 0) EXAMPLE 3 Multiply and simplify the product: x2 2 5x 1 6 3x? 2 4x 2 12 Solution x2 2 5x 1 6 3x? 2 4x 2 12 5 1 (x23) (x 2 2) 3x 1 2 (x 2 3) 1? 4 2 5 x 2 2 6x Answer x 2 2 6x (x 0, 3) EXERCISES Writing About ... |
2? a2 2 9 3 2s 1 4 21? 12 2a 2 6 (a 2 2)2 4b? 16b3 4 2 a2 y2 2 2y 2 3 2c3? 4c2 2y 1 2? 5x 1 15 x2 2 4 3x 3x 2 6 4x 1 8 6x 1 18 2 2 x 2x? d2 2 25 4 2 d2? 5d2 2 20 d 1 5 when x 65,908? 19. 22. 25. 28. 31. 34. 37. 40. 6. 11. 16. 5 d? d2 m2 32 8? 3m 3a 1 9 15a? 20. a3 18 7. 12. 17. x2 36? 20 6r2 10rs 5s2? 6r3 5x 2 5y x2y?... |
(2) Divide the numerators and denominators by the common factors: (3) Multiply the resulting numerators and the resulting denominators: Dividing Fractions 549 16c3 21d2 4 24c4 14d3 5 16c3 21d2? 14d3 24c4 2 5 16c3 21d2 3 2d? 14d3 24c4 3c 5 4d 9c Answer 4d 9c (c 0, d 0) EXAMPLE 2 Divide: 8x 1 24 x2 2 25 4 4x x2 1 8x 1 1... |
2x 4 2x 2 2 8x2 4a 4 (a2 2 b2) a2 2 ab 35 4 4b 12 7 7ab2 10cd 3y2 1 9y 4 14b3 5c2d2 18 4 16. 19. 5. 9. 8 4 x 2y xy2 x2y a3 2 a b 4 x y3 4 a3 4b3 6. 10. 14. 9 4 x x 3 6a2b2 8c 4 3ab x2 2 1 5 4 x 2 1 10 4 b2 2 4 b2 4 21x 3x 1 6 17. 20. b2 2 b 2 6 2b (x 2 2)2 4x2 2 16 13. 5y2 27 10 4 10a 1 15 25 4a2 2 9 12y 2 6 8 4 (2y2 ... |
ithmetic fractions Algebraic fractions Procedure To add (or subtract) fractions that have the same denominator: 1. Write a fraction whose numerator is the sum (or difference) of the numerators and whose denominator is the common denominator of the given fractions. 2. Reduce the resulting fraction to lowest terms. Add a... |
form that has the LCD as the, where x is the number by which the original denominator, we multiply by denominator must be multiplied to obtain the LCD 12 5 9 12 1 6 1 6 5 12 5 2 12 x x 2 2 B B A A Note that the LCD is the smallest possible common denominator. Procedure To add (or subtract) fractions that have differen... |
Inequalities Involving Fractions EXAMPLE 6 Solution Subtract: 6x x2 2 4 2 3 x 2 2 x2 4 (x 2)(x 2) x 2 (x 2) LCD (x 2)(x 2) 6x x2 2 4 2 3 x 2 2 3(x 1 2) (x 2 2)(x 1 2) 6x 5 (x 2 2)(x 1 2) 2 6x 2 3(x 1 2) 5 (x 2 2)(x 1 2) 5 6x 2 3x 2 6 (x 2 2)(x 1 2) 3x 2 6 (x 2 2)(x 1 2) 3(x 2 2) (x 2 2)(x 1 2) 5 5 5 3 x 1 2 Answer 3 x... |
7 5d b 2 3 5b 2 b 1 2 10b 4y 1 4 1x 8x 2 8 2 3x 4x 2 4 4 3x 1 18 x x2 2 36 x 2 5 2 x 2 x 1 3 22. 25. 28. 31. 34. 37. 40 3y 2 5 4y2 1 3y 5 2 2x x 1 y 2 3a 2 1 1 7 15a y2 2 9 y 2 3 1 2 1 2x 2 1 x 1 2 1 2x 2 3 23. 26. 29. 32. 35. 38. 41. 2 y 2 2 4 2 3c 2 3 6c2 7 2x 2 6 3y 2 4 5 3c 2 7 2c 5 x 2 3 1 10 3 2x 2 4 3x 2 6 1 6 ... |
. For the remainder of the trip, the car travels 2x 20 miles at 60 miles per hour. Represent the time that the car traveled at that speed in terms of x. c. Express, in terms of x, the total time for the two parts of the trip. 50. Ernesto walked 2 miles at a miles per hour and then walked 3 miles at (a 1) miles per hour... |
x 4 a. 4 5 20 1 x 3x 4 LCD 4 4 4 A A 5 4 20 1 x 4 B 5 4(20) 1 4 3x 4 3x 4 3x 5 80 1 x B B A x 4 A B 2x 5 80 x 5 40 Answer Solving Equations with Fractional Coefficients 557 b. 2x 1 7 6 2 2x 2 9 10 5 3 b. 30 2x 1 7 6 2 2x 2 9 10 5 3 LCD 30 B B 30 A 2x 1 7 6 2x 1 7 6 2 30 2 2x 2 9 10 2x 2 9 10 5 30(3) 5 30(3) B A A 5(2x ... |
the value of the nickels, 0.25(3x) the value of the quarters, and 0.10(x 5) the value of the dimes. Write the equation for the value of the coins. To simplify the equation, which contains coefficients that are decimal fractions with denominators of 100, multiply each side of the equation by 100. The total value of the... |
30 4 5 6 7 5 0 3 5 6x 2 9 3 1 x 10 26 6 2 t 2 25 t 2 3 5 5 4 21. 24. 0.4x 0.08 4.24 27. 1.7x 30 0.2x 18. 1 6t 5 18 m 2 2 9 5 3 5x 2 5 15 4 3y 1 1 4 5 44 2 y 5 4. 7. 10. 13. 16. 19. 3 2 r r 7y 12 2 1 3m 1 1 6 5 2 4 5 2y 2 5 3 4 5 2 2 3 2 2m 6 22. 25. 2c 0.5c 50 28. 0.02(x 5) 8 5. 3x 5 5 15 2r 1 6 8. 17. 11. 14. 5 5 24 ... |
In an isosceles triangle, each of the congruent sides is two-thirds of the base. The perimeter of the triangle is 42. Find the length of each side of the triangle. 45. The larger of two numbers is 12 less than 5 times the smaller. If the smaller number is equal to one-third of the larger number, find the numbers. 46. ... |
paca? 54. Sally spent half of her money on a present for her mother. Then she spent one-quarter of the cost of the present for her mother on a treat for herself. If Sally had $6.00 left after she bought her treat, how much money did she have originally? 55. Bob planted some lettuce seedlings in his garden. After a few ... |
500 less than the first sum, at 11% interest. If the annual interest on these two loans is $202.50, how much did he borrow at each rate? 562 Algebraic Fractions, and Equations and Inequalities Involving Fractions 14-7 SOLVING INEQUALITIES WITH FRACTIONAL COEFFICIENTS In our modern world, many problems involve inequalit... |
cents is greater than 200. 1 x 200 3x x 1 1 3x. 3(200) 3 A B 3x + x 600 4x 600 x 150 50 1 3x The number of cents that the younger boy has must be an integer greater than 50. The number of cents that the older boy has must be a multiple of 3 that is greater than 150. The younger boy has at least 51 cents. The older boy... |
than 40. Find the greatest possible integer. 27. The difference between three-fourths of a positive integer and one-half of that integer is greater than 28. Find the smallest possible integer. 28. The smaller of two integers is two-fifths of the larger, and their sum is less than 40. Find the largest possible integers... |
, what is the smallest amount he could have invested at 71 2%? 37. Mr. Lehtimaki wanted to sell his house. He advertised an asking price, but knew that he would accept, as a minimum, nine-tenths of the asking price. Mrs. Patel offered to buy the house, but her maximum offer was seven-eighths of the asking price. If the... |
6x B B 2x 1 6 5 3x 5 6x 6x 1 3 A B Answer x 6 EXAMPLE 2 Solve and check: 5x 1 10 x 1 2 5 7 Solution Multiply both sides of the equation by the least common denominator, x 2. 5x 1 10 x 1 2 5 7 (x 1 2) A 5x 1 10 x 1 2 5x 1 10 5 7x 1 14 5 (x 1 2)(7) B 22x 5 4 x 5 22 Check? 1 3 1 1 1 2 6 5? 1 2 1 2 5 1 2 ✔ Check 5x 1 10 x... |
2)(x 2 1) 2(x 2 1) 1 5 5 2(x 1 2)(x 2 1) 4x 1 2 5 x 1 2 3x 5 0 x 5 0 Answer x 0 Check? 2 1 1 1 21 5? 21 2 5 21 2 1 2(21) ✔ 1 2(0 2 1) 568 Algebraic Fractions, and Equations and Inequalities Involving Fractions EXERCISES Writing About Mathematics 1. Nathan said that the solution set of r 2 5 5 10 2 5r 2 25 agree with N... |
5 1 x 2x 2x 1 2x 1 1 3x 5 1 5x 3m 2 1 12y 8y 2b 1 1 5 b 2 2x 1 4 5 23 1 x 1 2 1 2x 1 2 3 In 46–49, solve each equation for x in terms of the other variables. Chapter Summary 569 46. t x 2 k 5 0 by c, y 5 c2 a x 5 47. t x 2 k 5 5k 48. a 1 b x 5 c 49, and c 0, is it possible to know the numerical value of x 50. If witho... |
defined only if values of the variables do not result in a denominator of 0. Fractions that are equal in value are called equivalent fractions. A fraction is reduced to lowest terms when an equivalent fraction is found such that its numerator and denominator have no common factor other than 1 or 1. This fraction is co... |
x 9. 10. 12. 13. 15. 2x 2 2 3x2 8 4? 9x 7b 4 18a 3a 35 3 1 ax ax 4 x2 2 5x x? x2 2x 2 10 x2 2 25 12 4 x2 2 10x 1 25 3y? 6 2 m 5m 6 5 xy 2 2 yz a 1 b 1 2b 2a a 1 b c 2 3 12 1 c 1 3 8 b 24. If the sides of a triangle are represented by, 2 perimeter of the triangle in simplest form. 16. 22. 21. 19. 18. 4 11. 14. 17. 20. 2... |
360 miles, find the rates at which Ross drove. (Express the time t 5 d needed for each part of the trip as r.) 572 Algebraic Fractions, and Equations and Inequalities Involving Fractions 39. The total cost, T, of n items that cost a dollars each is given by the equa- tion T na. a. Solve the equation T na for n in term... |
do you observe about the factors of the denominators? (9) Write a statement about terminating and infinitely repeating decimals based on your observations. CUMULATIVE REVIEW Part I Cumulative Review 573 CHAPTERS 1–14 Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will ... |
including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 574 Algebraic Fractions, and Equations and Inequalities Involving Fractions 11. Mrs. Kniger bought some stock on May 1 for $3,500. By Jun... |
biology, the study of genes inherited from one’s parents and grandparents is a direct application of probability. Probability helps us to predict when more tellers are needed at bank windows, when and where traffic jams are likely to occur, and what kind of weather we may expect for the next few days. While the list o... |
H) 5 1 2 P(tails) 5 1 2 or P(T) 5 1 2 Before we define probability, let us consider two more situations. 1. Suppose we toss a coin and it lands heads up. If we were to toss the coin a second time, would the coin land tails up? Is your answer “I don’t know”? Good! We cannot say that the coin must now be tails because we... |
results 1 where the probability was, or. This 2 fraction is called the relative frequency. Elizabeth had the lowest rel- 10 20 Albert Peter Thomas Maria Elizabeth Joanna Kathy Jeanne 8 13 12 10 6 12 11 7 6 20 ative frequency of heads,. Peter and Debbie tied for the highest relative. Maria’s relative frefrequency with ... |
. 6 20 for Peter and Debbie, the cumulative relative frequency, after While the relative frequency for individual students varied greatly, from for Elizabeth to 1 all 200 tosses were combined, was a number very close to. 2 A graph of the results of columns 4 and 5 will tell us even more. In the graph, the horizontal ax... |
coins, cards, and spinners will always be treated in this book as fair objects, unless otherwise noted. 2. Biased objects have been tampered with or are weighted to give one result a better chance of happening than another. The folded index card described earlier in this section is a biased object because the probabil... |
? Answers a. P(8) 5 1 10 b. P(odd) 5 10 or 1 2 EXAMPLE 4 A jar contains eight marbles: three are white and the remaining five are blue. In selecting a marble without looking, what is the probability it will be blue? (All marbles are the same size.) Answer P(blue) 5 5 8 Empirical Probability 581 EXAMPLE 5 The English al... |
tered tiles from a word game, a boy places 26 tiles, one tile for each letter of the alphabet, facedown on a table. After mixing up the tiles, he picks one. What is the probability that the tile contains one of the letters in the word MATH? 582 Probability 8. A tetrahedron is a four-sided object. Each side, or face, is... |
of the event. While you may wish to guess at the probability of the event before starting the experiment, conduct at least 100 trials to determine the best probability to be assigned. 11. An index card is folded in half and tossed. As described earlier in this section, the card may land in one of three positions: on i... |
, 15 had no absences for the year. What is the probability that a student, chosen at random, had no absences? 19. A chess club consists of 45 members of whom 24 are boys and 21 are girls. If a member of the club is chosen at random to represent the club at a tournament, what is the probability that the person chosen is... |
1. For Alma, the event of rolling a number greater than 4 contains only two outcomes: 5 and 6. 2. For Lee, a different event might be rolling a number less than 5. This event contains four outcomes: 1, 2, 3, and 4. Theoretical Probability 585 3. For Sandi, the event of rolling a 2 contains only one outcome: 2. When th... |
) ; P(2) P(1) We say that the die has uniform probability. If, however, a die is weighted to make it biased, then one or more sides will have a probability greater than while one or more sides will have a probabil- 1 6, ity less than theoretical probability does not apply to weighted objects.. A weighted die does not h... |
J event of selecting a jack. There are four jacks in the deck: jack of hearts, of diamonds, of spades, and of clubs. Thus, n(J) 4. P(J) number of possible jacks number of possible cards 5 n(J) n(S) 5 4 52 5 1 13 Answer Calculator Solution ENTER: 4 52 MATH 1 ENTER DISPLAY EXAMPLE 2 An aquarium at a pet store contains 8... |
Developing Skills 3. A fair coin is tossed. a. List the sample space. b. What is P(head), the probability that a head will appear? c. What is P(tail)? Theoretical Probability 589 In 4–9, a fair die is tossed. For each question: a. List the possible outcomes for the event. b. State the probability of the event. 4. The ... |
number 1,776 are written on disks and placed in a jar. What is the probabil- ity that the digit 7 will be chosen on a single random draw? 20. A letter is chosen at random from a given word. Find the probability that the letter is a vowel if the word is: a. APPLE b. BANANA c. GEOMETRY d. MATHEMATICS 590 Probability App... |
. In this example, event E rolling a 7; so, E { } or, and n(E) 0. The sample space S for rolling a die contains six possible outcomes, and n(S) 6. Therefore: P(E) number of ways to roll a 7 number of outcomes for the die 5 n(E) n(S) 5 0 6 5 0 In general, for any sample space S containing k possible outcomes, we say n(S... |
from a drawer containing only red sweaters. The Probability of Any Event The smallest possible probability is 0, for an impossible case; no probability can be less than 0. The largest possible probability is 1, for a certain event; no probability can be greater than 1. Many other events, as seen earlier, however, have... |
t least 10 cents? c. exactly 3 cents? d. more than 3 cents? Solution The sample space for this example is {N, D1, D2, Q}. Therefore, n(S) 4. a. There are two coins worth exactly 10 cents, D1 and D2. Therefore, n(E) 2 and P(coin worth 10 cents) n(E) n(S) 5 2 4 5 1 2. b. There are three coins worth at least 10 cents, D1,... |
arrow is spun to fall into one of the regions. For each question, find the probability that the arrow lands on the number described. 4. the number 5 6. a number less than 5 5. an even number 7. an odd number Evaluating Simple Probabilities 595 8. a number greater than 5 9. a number greater than 1 10. a number greater ... |
a question. Express, in decimal form, the probability that the student called upon is: a. a girl b. a boy c. a pupil in the class d. a person who is not a student in the class 596 Probability 22. The last digit of a telephone number can be any of the following: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Express, as a percent, t... |
BILITY OF (A AND B) If a fair die is rolled, we can find simple probabilities, since we know that S {1, 2, 3, 4, 5, 6}. For example, let event A be rolling an even number. Then A {2, 4, 6} and P(A) Let event B be rolling a number less than 3. Then B {1, 2} and n(S) 5 3 6 n(A) P(B) n(B) n(S) 5 2 6 Now, what is the proba... |
is rolled once. Find the probability of obtaining a number that is greater than 3 and less than 6. Solution Event A {numbers greater than 3} {4, 5, 6}. Event B {numbers less than 6} {1, 2, 3, 4, 5}. Event (A and B) {numbers greater than 3 and less than 6} {4, 5}. n(A d B) n(S) 5 2 6 Therefore: P(A and B) or P(A B) n(A... |
both science and math? b. A student from the class is selected at random. Find: (1) P(takes science) (2) P(takes math) (3) P(takes science and math) The Probability of (A or B) 599 7. At a karaoke party, some of the boys and girls take turns singing songs. Of the five boys, Patrick and Terence are teenagers while Bren... |
less than 3 on a die} {1, 2}. P(B) Then, event (A or B) {numbers that are even or less than 3}. n(B) n(S) 5 2 6 P(A or B) n(A or B) n(S) 5 4 6 600 Probability 5 3 6 5 2 6 Here P(A). In this case, the simple rule of addition does not work: P(A) P(B) P(A or B). What makes this example different from P(A or C), shown pre... |
is tossed. Event A {an even number} {2, 4, 6} Event B {a number less than 3} {1, 2} Event C {a number less than 2} {1} The Probability of (A or B) 601 We found that P(A or C) P(A) P(C) P(A or B) P(A) P(B) P(A and B) Why are these results different? Of these sets, A and C are disjoint, that is they have no element in c... |
is drawn at random. Find the probability that the card is: a. a king or an ace b. red or an ace Solution a. There are four kings in the deck, so P(king) 5 4 52. There are four aces in the deck, so P(ace) 5 4 52. These are mutually exclusive events. The set of kings and the set of aces are disjoint sets, having no elem... |
(Times) P(Chronicle) P(Times and Chronicle) P(Times and Chronicle) P(Chronicle) 5 3 10 5 1 5 5 2 1 3 10 1 6 5 2 3 10 10 2 3 10 5 8 10 or 4 5 Answer EXERCISES Writing About Mathematics 1. Let A and B be two events. Is it possible for P(A or B) to be less that P(A)? Explain why or why not. 2. If B is a subset of A, which... |
cents d. worth 10 cents or less e. worth 1 cent or more f. a quarter, a nickel, or a penny In 7–12, in each case choose the numeral preceding the expression that best completes the statement or answers the question. 7. If a single card is drawn from a standard deck, what is the probability that it is a 4 or a 9? (1) 2... |
a. the probability that a sophomore studies Spanish or French b. the number of sophomores who study one or more of these languages The Probability of (Not A) 605 15. The Greenspace Company offers lawn care services and snow plowing in the appropriate seasons. Of the 600 property owners in town, 120 have contracts for ... |
we draw a card from a standard deck, there are 52 sin. Since all singleton events are dis- gleton outcomes, each with a probability of joint, we can say: 1 52 P(king) P(king of hearts) P(king of diamonds) P(king of spades) P(king of clubs) 1 1 52 52 52 or 1 4 13 1 52 1 52 P(king) We also say: The sum of the probabilit... |
card is drawn. Find the probability that the card is: a. a heart d. not a picture card b. not a heart e. not an 8 c. a picture card f. not a red 6 g. not the queen of spades h. not an 8 or a 6 5. One letter is selected at random from the word PICNICKING. a. Find the probability of drawing each of the different letters... |
. Find the probability of drawing: (1) a quarter (2) a dime (3) a nickel b. Demonstrate that the sum of the three probabilities given as answers in part a is 1. c. Find the probability of not drawing: (1) a quarter (2) a dime (3) a nickel 13. The weather bureau predicted a 30% chance of rain. Express in fractional form... |
more activities is called a compound event. Before studying the probability of events based on two or more activities, let us study ways to count the number of elements or outcomes in a sample space for two or more activities. For example: A store offers five flavors of ice cream: vanilla, chocolate, strawberry, peach... |
space consists of 10 possible sundaes. Graph of ordered pairs Whether we use a tree diagram, a list of ordered pairs, or a graph of ordered pairs, we recognize that the sample space consists of 10 sundaes. The number of elements in the sample space can be found by multiplication: number of flavors number of toppings n... |
indicate any compound event of two or more activities. 2. Graphs should be limited to ordered pairs, or to events consisting of exactly two activities. EXAMPLE 1 The school cafeteria offers four types of salads, three types of beverages, and five types of desserts. If a lunch consists of one salad, one beverage, and o... |
Counting Principle for Probability: E and F are independent events. The probability of event E is m (0 m 1) and the probability of event F is n (0 n 1). The probability of the event in which E and F occur jointly is the product m n. Note 1: The product m n is within the range of values for a probability, namely, 0 # m... |
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