answers | investigation 2 · answers | investigation 2 pc. = -9,862.5 + 212.5t; to simplify p =...

Answers | Investigation 2

Applications

1. a. I = 12n + 150

b. E = 250 + 4.25n

c. +675; if you substitute 100 T-shirts into the income equation, you will get 12(100) + 150 = 1,350 in income, and if you substitute 100 into the expense equation, you will get E = 250 + 4.25(100) = 675. So, the profit is 1,350 - 675 = 675.

d. Possible answers: P = 12n + 150 - (250 + 4.25n), P = 12n + 150 - 4.25n - 250, or P = 7.75n - 100

2. a. E = 125n + 30n + 700 or E = 155n + 700

b. I = 350n

c. P = 350n - (125n + 30n + 700), P = 350n - 125n - 30n - 700, or P = 195n - 700

d. +4,175; substituting 25 for n into the profit equation, P = 195(25) - 700 = 4,175

e. 9 bikers; after substituting 1,055 for P in the equation P = 195n - 700 and solving for n, the equation is 1,055 = 195n - 700 or 1,755 = 195n. After dividing each side by 195, the number of bikers is 9.

f. Acceptable answers: The profit equation is a linear equation because it can be written in the form y = mx + b. It has a constant rate of change and a linear graph.

3. B

4. F

5. C

6. +375; since the probability of rain is 50, or 0.50, the number of predicted visitors is V = 600 - 500(0.50) = 350. Based on this number, the profit will be P = 2.50(350) - 500 = 375.

7. 72,; if students use the combined equation, which is P = -1,250R + 1,000 and solve for R when P = 100, they should get 72,. If students use both equations separately, then P = 10, and the number of visitors would be 240, which can be found by solving the equation 100 = 2.50n - 500 for n. So, to find the probability of rain, solve the other equation 240 = 600 - 500Rfor R to get 0.72 or 72,.

8. a. +325; combining both equations into one results in the equation B = 100 + 0.50(600 - 500R), or B = 400 - 250R. If the probability of rain is 30,, the daily employee-bonus fund is +325. Students may use both equations separately to find the number of visitors. V = 600 - 500R when R = 30,, which is 450. Then substitute 450 into the equation B = 100 + 0.50V and solve for V to get +325.

b. B = 100 + 0.50(600 - 500R) or B = 400 - 250R

c. +275; B = 400 - 250(0.5), which is B = +275

d. 10,; solving the equation 375 = 400 - 250R for R gives R = 0.1, or 10,.

9. a. 65°F; V = 50(T - 45) 1,000 = 50(T - 45) 1,000 = 50T - 2,250 1,000 + 2,250 = 50T - 2,250 + 2,250 3,250 = 50T

3,25050 = 50T

50 65 = T

b. To find the profit based on the temperature, substitute 50(T - 45) for V in the equation P = 4.25V - 300 to get P = 4.25[50(T - 45)] - 300.

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Investigation 2 Say It With Symbols 1


c. P = -9,862.5 + 212.5T; to simplify P = 4.25[50(T - 45)] - 300, first distribute the 50 by multiplying it by T and -45. Then multiply each of those terms by 4.25, and combine like terms:

P = 4.25[50(T - 45)] - 300 P = 4.25[50T - 2,250] - 300 P = -9,562.5 + 212.5T - 300 P = -9,862.5 + 212.5T

The 212.50 represents the rate of change for the profit when the temperature increases 1°. The y-intercept is -9,862.5. However, -9,862.5 does not have a physical meaning since T must always be greater than or equal to 458 to have a positive number of visitors. T represents the independent variable, or the temperature, and P represents the dependent variable, or the profit, which depends on the temperature because it changes at the rate of 212.50 dollars per 18° change in temperature.

d. +5,012.50; students may choose to use either of their equations from Question B, parts (b) or (c).

10. a. Consider the formula for volume of a cylinder, V = pr2h. With the radius doubled, V = p(2r)2h, which equals V = 4pr2h. So, when the radius is doubled, the volume increases by a factor of 4.

b. When the height is doubled, the volume increases by a factor of 2, V = pr2(2h) = 2pr2h.

11. a. Consider the formula for volume of a cone, V = 1

3pr2h. With the radius

doubled, V = 13p(2r)2h, which equals

V = 43pr2h. So, when the radius is

doubled, the volume increases by a factor of 4.

b. When the height is doubled, the volume increases by a factor of 2; V = 1

3pr2(2h) = 23pr2h.

12. a. Consider the formula for volume of a sphere, V = 4

3pr3. With the

radius doubled, V = 43p(2r)3, which

equals V = 43p

# 8r3 = 323 pr3. So,

when the radius is doubled, the volume increases by a factor of 8, or 23.

b. Similarly, when the radius is tripled, the volume increases by a factor of 27, or 33.

c. Similarly, when the radius is quadrupled, the volume increases by a factor of 64, or 43.

13. a. A = p * 52≈78.54 square feet

b. A = pr2

c. V = p * 52 * 3 + 46 * p * 53≈

497 cubic feet

d. V = pr2h + 46 * pr3

14. The volume of the cylindrical portion of the rocket plus the conical tip is V = pr2/ + 1

3pr2h.

15. The volume of the hemisphere plus the cylindrical body plus the conical end is V = 4

6pr3 + pr2/ + 13pr2h.

16. a. The volume of the prism is 6 * 6 *4 cubic units = 144 cubic units. So, the volume of the pyramid is 1443 = 48 cubic units.

b. A cube with edge length 2 units would have volume 8 cubic units. A pyramid that fits inside of this cube would have the given volume.

c. A cube with edge length 3 units would have volume 27 cubic units. A pyramid that fits inside this cube would have the given volume.

d. A cube with edge length 3x units would have volume 27x3 cubic units. So, a pyramid with base 3x by 3x and height 3x units would have a volume of 13(27x3) = 9x3.


Investigation 2Say It With Symbols 2


Connections

17. J; students can try an example like a = 1 and b = 2 to check that J is false. The other letters are true; F and H are the Associative Property of Addition and Multiplication, respectively, and G is the Commutative Property of Multiplication.

18. x(x + 5) = x2 + 5x

x 5

x2 5xx

19. (2 + x)(2 + 3x) = (2 + x)2 + (2 + x)3x =4 + 2x + 6x + 3x2 = 4 + 8x + 3x2

3x22xx

6x4

3x2

2

20. (x + 2)(2x + 3) = (x + 2)2x + (x + 2)3 = 2x2 + 4x + 3x + 6 = 2x2 + 7x + 6

2x2

2x

4x2 6

3xx

3

21. a. 11x - 12 = 30 + 5x 11x - 12 + 12 = 30 + 12 + 5x 11x = 42 + 5x 11x - 5x = 42 + 5x - 5x 6x = 42

6x6 = 42

6 x = 7

b. To check, substitute 7 into the original equation for x and see if the values on each side of the equal sign are equal to each other.

11x - 12 = 30 + 5x 11(7) - 12 = 30 + 5(7) 77 - 12 = 30 + 35 65 = 65

c. To solve the equation using a graph, first graph each of the equations y = 11x - 12 and y = 30 + 5x and use the x-value of their point of intersection for the solution. To solve the equation using a table, look on the tables for each equation and see for which value of x their y-values are equal.

22. a. +1,000

b.

500

1,000

3,000

Number ofSoccer Balls

500

1,000

3,000

Income FromSoccer Balls

(dollars)

1,250

1,500

2,500

Expenses ofSoccer Balls

(dollars)

c. To find the profit of soccer balls, subtract the expenses from the income. See table below.

500

1,000

3,000


500 – 1,250 = –750

1,000 – 1,500 = –500

3,000 – 2,500 = 500

Profit of Soccer Balls(dollars)

d. The break-even point is at 2,000 soccer balls; the income and expenses are both +2,000.

e. Income = 1 times the number of soccer balls or I = 1n

Expenses = 1,000 + 0.5 times number of soccer balls or E = 1,000 + 0.5n

Profit = Income - Expenses, P = 1n - (1,000 + 0.5n), P = n - 1,000 - 0.5n, or P = 0.5x - 1,000




f. + -125 or a loss of 125 dollars (See Figure 1.)

g. +22,000; Profit = -1,000 + 0.5 (number of

soccer balls) +10,000 = -1,000 + 0.5n 10,000 + 1,000 = -1,000 + 1,000 + 0.5n 11,000 = 0.5n

11,0000.5 = 0.5n

0.5 22,000 = n The number of soccer balls produced

and sold if the profit is +10,000 is 22,000.

23. One possible solution:

7x + 15 = 12x + 5 7x - 7x + 15 = 12x - 7x + 5 15 = 5x + 5 15 - 5 = 5x + 5 - 5 10 = 5x 10

5 = 5x5

2 = x

Check:

7x + 15 = 12x + 5 7(2) + 15 = 12(2) + 5 14 + 15 = 24 + 5 29 = 29

24. x = 2; the solution is the same as Exercise 23 because the Commutative Property does not change the value of the variables when you are solving an equation.


-3x + 5 = 2x - 10 -3x - 2x + 5 = 2x - 2x - 10 -5x + 5 = -10 -5x + 5 - 5 = -10 - 5 -5x = -15

-5x-5 = -15

-5 x = 3

Check:

-3x + 5 = 2x - 10 -3(3) + 5 = 2(3) - 10 -9 + 5 = 6 - 10 -4 = -4

26. One possible method:

14 - 3x = 1.5x + 5 14 - 3x - 14 = 1.5x + 5 - 14 -3x = 1.5x - 9 -3x - 1.5x = 1.5x - 9 - 1.5x ( -3 - 1.5)x = -9 -4.5x = -9 x = 2

Check:

14 - 3(2) = 1.5(2) + 5 8 = 3 + 5 8 = 8


9 - 4x = (3 + x)2

2(9 - 4x) = 2 # (3 + x)2

18 - 8x = 3 + x 18 - 8x - x = 3 + x - x 18 - 9x = 3 18 - 9x - 18 = 3 - 18 -9x = -15

-9x-9 = -15

-9

x = 123

Check:

9 - 4(123) =

3 + (123)

2

9 - 623 =

423

2

213 = 21

3

Figure 1

1,750 1,750


Income = # ofSoccer Balls

l = 1n

Expenses = 1,000 +0.5 (# of soccer balls)

Profit = Income − Expensesor P = 0.5n − 1,000

1,000 + 0.5(1,750) =1,000 + 875 = 1,875

1,750 – 1,875 = –125, or0.5(1,750) – 1,000 =

875 – 1,000 = –125





-3(x + 5) = (2x - 10)3

-3x - 15 = (2x - 10)3

3( -3x - 15) = 3 # (2x - 10)3

-9x - 45 = 2x - 10 -9x - 2x - 45 = 2x - 2x - 10 -11x - 45 = -10 -11x - 45 + 45 = -10 + 45 -11x = 35

-11x-11 = 35

-11

x = -3 211

Check:

-3( -3 211 + 5) =

2( -3 211) - 10

3

-3( -1 911) =

-7011 - 10

3

-6011 =

-180113

-6011 = -60

11

29. a. The two bids are equal when the y-values for a common x-value are equal. This occurs when x = 25 and y = 200, meaning the bids are both +200 for 25 books.

10

15

20

25

30

x(Number of

books printed)

140

160

180

200

220

95

130

165

200

235

y = 100 + 4x(dollars)

y = 25 + 7x(dollars)

b. y

x

200250300350400

100150

500

0 10 20 30 40 50

y = 25 + 7x, Bid 2

y = 100 + 4x, Bid 1

(25, 200)

c. For 25 books, the bids are equal. The graph shows that for more than 25 books, Bid 1 is less than Bid 2 because the graph for Bid 1 is below the graph for Bid 2. For example, if the number of books is 26, Bid 1 is +204 and Bid 2 is +207. Since Bid 2 increases more for each book, if the number of books is greater than 25, Bid 1 is lower.

30. a. Bid 1: 100 + 4(75) = +400, Bid 2: 25 + 7(75) = +550

Students might also find these values from the table or graph.

b. The greatest number of books that can be printed is 50 for Bid 1 and 39 for Bid 2. Explanations will vary. Students might extend their tables or graphs, use trial-and-error, or apply methods for solving linear equations.

31. The related equations are y = 8x and y = 30 + 6x. The two bids are equal when x = 15 and y = 120, meaning they are both +120 for 15 books. Explanations will vary: students may use tables, graphs, or begin to see a pattern and solve the equations 8x = 30 + 6x symbolically.

32. 3.14 * 4.252 * 5.5 represents the volume of a cylinder with radius 4.25 and height 5.5.

33. 13 * 3.14 * 4.252 * 5.5 represents the volume of a cone with radius 4.25 and height 5.5.

34. 43 * 3.14 * 4.253 represents the volume of a sphere with radius 4.25.

35. 13 * 3.14 * 4.252 * 5.5 + 4

6 * 3.14 * 4.253 represents the volume of a cone with radius 4.25 and height 5.5 and a hemisphere with a radius of 4.25.




36. 3.14 * 4.252 * 5.5 + 46 * 3.14 * 4.253

represents the volume of a cylinder with radius 4.25 and height 5.5 and a hemisphere with a radius of 4.25.

37. 3.14 * 4.252 * 5.5 + 13 * 3.14 * 4.252 * 3.5

represents the volume of a cylinder with radius 4.25 and height 5.5 combined with a cone with a radius of 4.25 and a height of 3.5.

38. 3.14 * 4.252 * 5.5 - 46 * 3.14 * 4.253

represents the volume of a cylinder with radius 4.25 and height 5.5 with a hollow space of a hemisphere with a radius of 4.25 carved out.

39. 3.14 * 4.252 * 5.5 - 13 * 3.14 * 4.252 * 3.5

represents the volume of a cylinder with radius 4.25 and height 5.5 with a hollow space of a cone with a radius of 4.25 and height of 3.5 carved out.

Extensions

40. a. P = 15x - 500 + 106x - x2

b. The maximum profit is +3,160.25, which occurs between 60 and 61 people. That is, the maximum profit occurs when x = 60.5, which means 6,050 people since x is in hundreds.

41. a. The vertical slice of a cylinder results in a rectangular face.

b. The horizontal slice of a cylinder results in a circular face.

42. a. The vertical slice of a cone results in a triangular face.

b. The horizontal slice of a cone results in a circular face.

43. a. The vertical slice of a sphere results in a circular face.

b. The horizontal slice of a sphere results in a circular face.



answers | investigation 2 · answers | investigation 2 pc. = -9,862.5 + 212.5t; to simplify p =...

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