SECTION 3.1 Properties of Linear Functions and Linear Models 147 (a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity? (b) Determine the prices for which quantity demanded is less than quantity supplied. (c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied? 43. Taxes The function T x x 0.12 11,600 1160 ( ) ( ) = − + represents the 2024 tax bill T for a person whose filing status is an unmarried individual and whose taxable income is x dollars for income over $11,600 but not over $47,150. Source: Internal Revenue Service. (a) What is the domain of this linear function? (b) What is this person’s tax bill if the taxable income is $20,000? (c) Which variable is independent? Which is dependent? (d) Graph the linear function over the domain specified in part (a). (e) What is this person’s taxable income if the tax bill is $3143? 44. Competitive Balance Tax Instead of a salary cap for teams, under the 2022–2026 labor agreement between Major League Baseball and the players, any team whose payroll exceeded $230 million in 2022 had to pay a competitive balance tax of 20% on all overages. The linear function T p p 0.20 230 ( ) ( ) = − describes the competitive balance tax T for a team whose payroll was p (in millions of dollars). Source: Major League Baseball. (a) What is the domain of this linear function? (b) What was the competitive balance tax for the New York Mets, whose 2022 payroll was $288 million? (c) Graph the linear function. (d) What was the 2022 payroll for the Los Angeles Dodgers, who paid a competitive balance tax of $6.7 million? The point at which a company’s profits equal zero is called the company’s break-even point. For Problems 45 and 46, let R represent a company’s revenue, let C represent the company’s costs, and let x represent the number of units produced and sold each day. (a) Find the firm’s break-even point; that is, find x so that R C. = (b) Solve the inequality R x C x ( ) ( ) > to find the units that represent a profit for the company. 45. R x x C x x 8 4.5 17,500 ( ) ( ) = = + 46. R x x C x x 12 10 15,000 ( ) ( ) = = + 47. Straight-line Depreciation Suppose that a company has just purchased a new computer for $3000. The company chooses to depreciate the computer using the straight-line method over 3 years. (a) Write a linear model that expresses the book value V of the computer 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 computer after 2 years? (e) When will the computer have a book value of $2000? 38. Phone Charges The monthly cost C, in dollars, for calls from the United States to Germany on a certain wireless plan is modeled by the function C x x 0.26 5, ( ) = + where x is the number of minutes used. (a) What is the cost if you talk on the phone for 50 minutes? (b) Suppose that your monthly bill is $21.64. How many minutes did you use the phone? (c) Suppose that you budget $50 per month for calls to Germany.What is the maximum number of minutes that you can talk? (d) What is the domain of C if there are 30 days in the month? 39. Sleep In a study looking at total amount of sleep by adolescents 15 years of age, researchers determined there was a linear relation between school start times and total amount of daily sleep. Let x represent the number of minutes from 6 a.m. that school starts, and let S represent the total number of minutes slept. The model ( ) = + S x x 0.35 400 describes the relation between total sleep and school start time. Source: Nahmod, N. G., Lee, S., Master, L., Chang, A. M., Hale, L., Buxton, O. M. “Later high school start times associated with longer actigraphic sleep duration in adolescents.” Sleep. 2019; 42(2): zsy212. doi:10.1093/sleep/zsy212 (a) What is the total number of minutes slept if school starts at 6:30 am? (b) Suppose the total number of minutes slept is 440. To the nearest minute, what time does school start? 40. CO2 and Energy Production A model that relates the carbon dioxide emissions (in millions of tons), C, and electricity produced (in terawatt-hours), x, is given by the linear function C x x 1.3491 11.2847 ( ) = − for all countries in the world. Source: www.bp.com (a) What is the carbon dioxide emissions for a country that produces 1500 terawatt-hours of electricity? Round your answer to the nearest whole number. (b) If a country emits 3000 million tons of carbon dioxide, how much electricity does it produce? Round your answer to the nearest whole number. 41. Supply and Demand Suppose that the quantity supplied S and the quantity demanded D of T-shirts at a concert are given by the following functions: S p p D p p 600 50 1200 25 ( ) ( ) = − + = − where p is the price of a T-shirt. (a) Find the equilibrium price for T-shirts at this concert. What is the equilibrium quantity? (b) Determine the prices for which quantity demanded is greater than quantity supplied. (c) What do you think will eventually happen to the price of T-shirts if quantity demanded is greater than quantity supplied? 42. Supply and Demand Suppose that the quantity supplied S and the quantity demanded D of hot dogs at a baseball game are given by the following functions: S p p D p p 2000 3000 10,000 1000 ( ) ( ) = − + = − where p is the price of a hot dog.
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