148 CHAPTER 3 Linear and Quadratic Functions 52. Hooke’s Law The distance d between the bottom of a suspended spring and a countertop is a linear function of the weight w attached to the bottom of the spring.The bottom of the spring is 9 inches from the countertop when the attached weight is 1.5 pounds and 5 inches from the countertop when the attached weight is 2.5 pounds. (a) Find a linear model that relates the distance d from the countertop and the weight w. (b) Find the distance between the bottom of the spring and the countertop if no weight is attached. (c) What is the smallest weight that will make the bottom of the spring reach the countertop? (Ignore the thickness of the weight.) 53. Mixed Practice Building a Linear Model from Data The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with $60. Soda, s Hot Dogs, h 20 0 15 3 10 6 5 9 (a) Plot the ordered pairs s h , ( ) in a Cartesian plane. (b) Show that the number h of hot dogs purchased is a linear function of the number s of sodas purchased. (c) Determine the linear function that describes the relation between s and h. (d) What is the domain of the linear function? (e) Graph the linear function in the Cartesian plane drawn in part (a). (f) Interpret the slope. (g) Interpret the intercepts. 54. Challenge Problem Temperature Conversion The linear function F C C 9 5 32 ( ) = + converts degrees Celsius to degrees Fahrenheit, and the linear function R F F 459.67 ( ) = + converts degrees Fahrenheit to degrees Rankine. Find a linear function that converts degrees Rankine to degrees Celsius. 48. Straight-line Depreciation Suppose that a company has just purchased a new machine for its manufacturing facility for $120,000. The company chooses to depreciate the machine using the straight-line method over 10 years. (a) Write a linear model that expresses the book value V of the machine as a function of its age x. (b) What is the domain of the function found in part (a)? (c) Graph the linear function. (d) What is the book value of the machine after 4 years? (e) When will the machine have a book value of $72,000? 49. Cost Function The simplest cost function C is a linear cost function, C x mx b, ( ) = + where the y-intercept b represents the fixed costs of operating a business and the slope m represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of $1800, and each bicycle costs $90 to manufacture. (a) Write a linear model that expresses the cost C of manufacturing x bicycles in a day. (b) Graph the model. (c) What is the cost of manufacturing 14 bicycles in a day? (d) How many bicycles could be manufactured for $3780? 50. Cost Function Refer to Problem 49. Suppose that the landlord of the building increases the bicycle manufacturer’s rent by $100 per month. (a) Assuming that the manufacturer is open for business 20 days per month, what are the new daily fixed costs? (b) Write a linear model that expresses the cost C of manufacturing x bicycles in a day with the higher rent. (c) Graph the model. (d) What is the cost of manufacturing 14 bicycles in a day? (e) How many bicycles can be manufactured for $3780? 51. Hooke’s Law According to Hooke’s Law, a linear relationship exists between the distance that a spring stretches and the force stretching it. Suppose a weight of 0.5 kilogram causes a spring to stretch 2.75 centimeters and a weight of 1.2 kilograms causes the same spring to stretch 6.6 centimeters. (a) Find a linear model that relates the distance d of the stretch and the weight w. (b) What stretch is caused by a weight of 2.4 kilograms? (c) What weight causes a stretch of 19.8 centimeters? Explaining Concepts 55. Which functions might have the graph shown? (More than one answer is possible.) (a) f x x2 7 ( ) = − (b) g x x3 4 ( ) = − + (c) H x 5 ( ) = (d) F x x3 4 ( ) = + (e) G x x 1 2 2 ( ) = + 56. Which functions might have the graph shown? (More than one answer is possible.) (a) f x x3 1 ( ) = + (b) g x x2 3 ( ) = − + (c) H x 3 ( ) = (d) F x x4 1 ( ) = − − (e) G x x 2 3 3 ( ) = − + y x y x 57. Under what circumstances is a linear function f x mx b ( ) = + odd? Can a linear function ever be even? 58. Explain how the graph of f x mx b ( ) = + can be used to solve mx b 0. + >
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