Answers Investigation 2

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1 Answers Investigation Applications 1. a. I = 1n b. E = n c. +675; if you substitute 100 T-shirts into the income equation, you will get 1(100) = 1,50 in income, and if you substitute 100 into the epense equation, you will get E = (100) = 675. So, the profit is 1, = 675. d. Possible answers: P = 1n ( n), P = 1n n - 50, or P = 7.75n a. E = 15n + 0n or E = 155n B 4. F 5. C b. I = 50n c. P = 50n - (15n + 0n + 700), P = 50n - 15n - 0n - 700, or P = 195n d. +4,175; substituting 5 for n into the profit equation, P = 195(5) = 4,175 e. 9 bikers; after substituting 1,055 for P in the equation P = 195n and solving for n, the equation is 1,055 = 195n 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 = m + b. It has a constant rate of change and a linear graph ; since the probability of rain is 50, or 0.50, the number of predicted visitors is V = (0.50) = 50. Based on this number, the profit will be P =.50(50) = ,; if students use the combined equation, which is P = -1,50R + 1,000 and solve for R when P = 100, they should get 7,. If students use both equations separately, then P = 10, and the number of visitors would be 40, which can be found by solving the equation 100 =.50n for n. So, to find the probability of rain, solve the other equation 40 = R for R to get 0.7 or 7,. 8. a. +5; combining both equations into one results in the equation B = ( R), or B = R. If the probability of rain is 0,, the daily employeebonus fund is +5. Students may use both equations separately to find the number of visitors. V = R when R = 0,, which is 450. Then substitute 450 into the equation B = V and solve for V to get +5. b. B = ( R) or B = R c. +75; B = (0.5), which is B = +75 d. 10,; solving the equation 75 = R for R gives R = 0.1, or 10,. 9. a. 65 F; V = 50(T - 45) 1,000 = 50(T - 45) 1,000 = 50T -,50 1,000 +,50 = 50T -,50 +,50,50 = 50T,50 50 = 50T = T b. To find the profit based on the temperature, substitute 50(T - 45) for V in the equation P = 4.5V - 00 to get P = 4.5[50(T - 45)] Say It With Symbols 1 Investigation

2 Answers Investigation c. P = -9, T; to simplify P = 4.5[50(T - 45)] - 00, first distribute the 50 by multiplying it by T and -45. Then multiply each of those terms by 4.5, and combine like terms: P = 4.5[50(T - 45)] - 00 P = 4.5[50T -,50] - 00 P = -9, T - 00 P = -9, T The 1.50 represents the rate of change for the profit when the temperature increases 1. The y-intercept is -9,86.5. However, -9,86.5 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 1.50 dollars per 18 change in temperature. d. +5,01.50; 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 = pr h. With the radius doubled, V = p(r) h, which equals V = 4pr h. 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, V = pr (h) = pr h. 11. a. Consider the formula for volume of a cone, V = 1 pr h. With the radius doubled, V = 1 p(r) h, which equals V = 4 pr h. 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 ; V = 1 pr (h) = pr h. 1. a. Consider the formula for volume of a sphere, V = 4 pr. With the radius doubled, V = 4 p(r), which equals V = 4 # p 8r = pr. So, when the radius is doubled, the volume increases by a factor of 8, or. b. Similarly, when the radius is tripled, the volume increases by a factor of 7, or. c. Similarly, when the radius is quadrupled, the volume increases by a factor of 64, or a. A = p * square feet b. A = pr c. V = p * 5 * * p * cubic feet d. V = pr h * pr 14. The volume of the cylindrical portion of the rocket plus the conical tip is V = pr / + 1 pr h. 15. The volume of the hemisphere plus the cylindrical body plus the conical end is V = 4 6 pr + pr / + 1 pr h. 16. a. The volume of the prism is 6 * 6 * 4 cubic units = 144 cubic units. So, the volume of the pyramid is 144 = 48 cubic units. b. A cube with edge length 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 units would have volume 7 cubic units. A pyramid that fits inside this cube would have the given volume. d. A cube with edge length units would have volume 7 cubic units. So, a pyramid with base by and height units would have a volume of 1 (7 ) = 9. Say It With Symbols Investigation

3 Answers Investigation Connections 17. J; students can try an eample like a = 1 and b = 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. ( + 5) = ( + )( + ) = ( + ) + ( + ) = = ( + )( + ) = ( + ) + ( + ) = = a = = = = = = 4 6 = 7 Say It With Symbols b. To check, substitute 7 into the original equation for and see if the values on each side of the equal sign are equal to each other = (7) - 1 = 0 + 5(7) 77-1 = = 65 c. To solve the equation using a graph, first graph each of the equations y = 11-1 and y = and use the -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 their y-values are equal.. a. +1,000 b. Number of 500 1,000,000 Income From 500 1,000,000 Epenses of 1,50 1,500,500 c. To find the profit of soccer balls, subtract the epenses from the income. See table below. Number of 500 1,000,000 Profit of 500 1,50 = 750 1,000 1,500 = 500,000,500 = 500 d. The break-even point is at,000 soccer balls; the income and epenses are both +,000. e. Income = 1 times the number of soccer balls or I = 1n Epenses = 1, times number of soccer balls or E = 1, n Profit = Income - Epenses, P = 1n - (1, n), P = n - 1, n, or P = 0.5-1,000 Investigation

4 Answers Investigation f or a loss of 15 dollars (See Figure 1.) g. +,000; Profit = -1, (number of soccer balls) +10,000 = -1, n 10, ,000 = -1, , n 11,000 = 0.5n 11,000 = 0.5n ,000 = n The number of soccer balls produced and sold if the profit is +10,000 is,000.. One possible solution: = = = = = = 5 5 = = () + 15 = 1() = = 9 4. = ; the solution is the same as Eercise because the Commutative Property does not change the value of the variables when you are solving an equation. 5. One possible solution: = = = = = = = = () + 5 = () = = One possible method: 14 - = = = = ( ) = = -9 = 14 - () = 1.5() = = 8 7. One possible solution: ( + ) 9-4 = (9-4) = # ( + ) 18-8 = = = = = = = 1 9-4(1 ) = + (1 ) 9-6 = 4 1 = 1 Figure 1 Number of Income = # of l = 1n 1,750 1,750 Epenses = 1, (# of soccer balls) 1, (1,750) = 1, = 1,875 Profit = Income Epenses or P = 0.5n 1,000 1,750 1,875 = 15, or 0.5(1,750) 1,000 = 875 1,000 = 15 Say It With Symbols 4 Investigation

5 Answers Investigation 8. One possible solution: ( - 10) -( + 5) = ( - 10) = ( ) = # ( - 10) = = = = = = = - 11 ( -( ) ) = -70 -( -1 11) 9 11 = = = a. The two bids are equal when the y-values for a common -value are equal. This occurs when = 5 and y = 00, meaning the bids are both +00 for 5 books. (Number of books printed) y = y = b. y (5, 00) y = 5 + 7, Bid y = , Bid 1 c. For 5 books, the bids are equal. The graph shows that for more than 5 books, Bid 1 is less than Bid because the graph for Bid 1 is below the graph for Bid. For eample, if the number of books is 6, Bid 1 is +04 and Bid is +07. Since Bid increases more for each book, if the number of books is greater than 5, Bid 1 is lower. 0. a. Bid 1: (75) = +400, Bid : 5 + 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 9 for Bid. Eplanations will vary. Students might etend their tables or graphs, use trial-and-error, or apply methods for solving linear equations. 1. The related equations are y = 8 and y = The two bids are equal when = 15 and y = 10, meaning they are both +10 for 15 books. Eplanations will vary: students may use tables, graphs, or begin to see a pattern and solve the equations 8 = symbolically...14 * 4.5 * 5.5 represents the volume of a cylinder with radius 4.5 and height *.14 * 4.5 * 5.5 represents the volume of a cone with radius 4.5 and height *.14 * 4.5 represents the volume of a sphere with radius *.14 * 4.5 * *.14 * 4.5 represents the volume of a cone with radius 4.5 and height 5.5 and a hemisphere with a radius of 4.5. Say It With Symbols 5 Investigation

6 Answers Investigation * 4.5 * *.14 * 4.5 represents the volume of a cylinder with radius 4.5 and height 5.5 and a hemisphere with a radius of * 4.5 * *.14 * 4.5 *.5 represents the volume of a cylinder with radius 4.5 and height 5.5 combined with a cone with a radius of 4.5 and a height of * 4.5 * *.14 * 4.5 represents the volume of a cylinder with radius 4.5 and height 5.5 with a hollow space of a hemisphere with a radius of 4.5 carved out * 4.5 * *.14 * 4.5 *.5 represents the volume of a cylinder with radius 4.5 and height 5.5 with a hollow space of a cone with a radius of 4.5 and height of.5 carved out. Etensions 40. a. P = b. The maimum profit is +,160.5, which occurs between 60 and 61 people. That is, the maimum profit occurs when = 60.5, which means 6,050 people since 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. 4. a. The vertical slice of a cone results in a triangular face. b. The horizontal slice of a cone results in a circular face. 4. a. The vertical slice of a sphere results in a circular face. b. The horizontal slice of a sphere results in a circular face. Say It With Symbols 6 Investigation