Answers Investigation 3

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1 Answers Investigation Applications 1. a. (See Figure 1.) b. Rectangles With Area 1 in. b. Points will var. Sample: Rectangles With Area in. 1 1 Width (in.) 1 Width (in.) 1 Length (in.) c. As length increases, width decreases at a decreasing rate. d. w = 1 1 / (or w/ = 1, or / = w ); not linear. a. Values in table will var. Sample: (See Figure.) Length (in.) c. w = / ; not linear d. The graphs are similar in shape, but the coordinates of the points are different. e. The equations have the same form, but the constant is different. Figure 1 Rectangles With Area 1 in. Length (in.) Width (in.) 1 Figure Rectangles With Area in. Length (in.) 1 1 Width (in.) 1 Thinking With Mathematical Models 1 Investigation

2 Answers Investigation. Analzing breaking weight data. a. Answers will var, but =, where is the length and is the breaking weight, is a reasonable choice. b. In the equation =, (or length) is in the denominator, so as increases, (or breaking weight) decreases. This is reasonable because the data show that as the length of a bridge increases, the strength decreases.. not an inverse variation. inverse variation; =. inverse variation; = 1. not an inverse variation. a. Answers will var, but should fit the patterns in the table and graph below. Marathon Speeds 9. a. c. decreases b. mi/h; decreases b.1 mi/h; decreases b 1.1 mi/h d. Equal changes in time do not alwas result in equal changes in average speed. Vehicle Tpe Large Truck Large SUV Limousine Large Sedan Small Truck Sports Car Compact Car Sub-Compact Car Fuel Used (gal) Fuel Efficienc (mi/gal) Time (h) Running Speed (mi/h) b. Fuel Efficienc Average Speed (mi/h) Marathon Speeds 1. a. Fuel Efficienc (mi/gal) Fuel Used (gal) c. e =, f, where f is the amount of fuel used d. decreases b mpg; decreases b. mpg; decreases b. mpg e. Equal changes in fuel use do not alwas lead to equal changes in fuel efficienc. Time (h) Charit Bike Ride Riding Speed (mi/h) 1.. Time (h) 1.1 b. s =. t 1.1. Thinking With Mathematical Models Investigation

3 Answers Investigation Riding Time (h) 1 Charit Bike Ride 1 1 Average Speed (mi/h) b. t =, s, where s is the speed in miles per hour. c. decreases b. hours; decreases b. hours; decreases b 1. hours d. For constant change in average riding speed, the change in time is not constant. Connections 1. a. slope positive, -intercept, passes through the origin at (, ) b. slope positive, -intercept positive, crosses the -ais to the left of the origin c. slope, -intercept negative, never crosses the -ais d. slope negative, -intercept positive, crosses the -ais to the right of the origin 11. a. (See Figure.) 1. a. Points per Question 1 Mr. Einstein s Tests 1 1 Number of Questions b. p = 1 n c. decreases b points per question; decreases b. points per question; decreases b.1 points per question; decreases b. points per question d. For constant change in the number of questions, the change in points per question is not constant. e. slope negative, -intercept negative, crosses the -ais to the left of the origin =. + O ] Figure Thinking With Mathematical Models Investigation

4 Answers Investigation b. c. d. = + O = + O = O b Width (ft) 1 1 Rectangles With Perimeter ft 1 1 Length (ft) c. As length increases, width decreases. The rate of change is constant. - / d. w = - / or w =. This function is linear. The graph is a straight line and the equation has the form = m + b a. (See Figure.) A number and its additive inverse are the same distance from on the number line. The labeled number line has reflection smmetr. (See Figure.) Figure Rectangles With Perimeter ft Length (ft) 1 1 Width (ft) 1 1 Figure Thinking With Mathematical Models Investigation

5 Answers Investigation Numbers greater than 1 have multiplicative inverses between and 1. Numbers less than -1 have multiplicative inverses between -1 and. (See Figure.) 9. C. a.. b.. 1. = c. because the relationship between number of quizzes and average quiz scores is not linear - =- = = To solve with a graph, graph = - and find the -coordinate of the point where =-. To solve with a table, make a table of (, ) values for = -, find the -value corresponding to =-.. = 1 - = = 1-1 = 1 = To solve with a graph, graph = 1 - and = - 1, and find the -coordinate of the intersection point. To solve with a table, make a table of (, ) values for = - 1 and = 1 -. Then find the -value for which the -values for the two equations are the same.. = 1 +. = -. = = 1 +. To find the slope, take the points (, 1) and (, ) on the line and find the vertical change () and the horizontal change (). Slope is the ratio rise run = = 1.. a. A = n (Problem. Question D) and d = t (Problem. Question C). The ratio is and the ratio is. b. The ratio equals k in both cases. c. changes b k as changes b 1. This pattern results in a straight-line graph with slope k. d. With a direct variation the graph is a line, and the equation is of the form = k where the slope of the line equals k. With an inverse variation, the graph is a curve, and the equation is of the form = k.. Super Market charges about +. per tomato and Gus s Groceries charges about +... Super Market; Gus s Groceries charges about +. per cucumber and Super Market charges +... Gus s Groceries; Gus s Groceries charges +. per apple and Super Market charges about a. about +.; +.,.. b. about +.;. * 1. c. about.n (or eactl 1 n ) Figure Thinking With Mathematical Models Investigation

6 Answers Investigation Etensions. a. If is the number of tickets sold and is the profit, then = b. Spring Show Ticket Sales Tickets Sold Total Profit Per-Ticket Profit $ $ $ $ $9 $1, $1, $1, $1, $,1 $1. $. $. $. $.9 $. $. $.1 $.1 $. c. Total Profit ($), 1, 1, 1, Spring Show Ticket Sales 1 Tickets Sold e. Spring Show Ticket Sales Per-Ticket Profit ($) 1 1 Ticket Sales f. The pattern for total profit is linear; the pattern for per-ticket profit is not. The graph for total profit is a straight line; the graph for per-ticket profit is a curve. In the column for total profit, there is a constant difference in values; in the column for per-ticket profit, there is not. The per-ticket profit increases b smaller and smaller amounts as the number of tickets sold increases.. a. cm b. 1 cm b 1 cm b. cm c. The surface area of the original prism is cm. The surface area of the prism in part (b) is cm. The surface area of the original prism is smaller.. Ms. Singh traveled mi in hr, for an average speed of mi/h. mi/h.. (, 1), (1, ); 1c =, c =.. (, 9),(, ); c =, c =. (, ), (, ); c = 1, c =.. A 9. G. A d. See part (b) above. Thinking With Mathematical Models Investigation