76 CHAPTER 2 Functions and Their Graphs 115. Health Care Suppose that ( ) P x represents the percentage of income spent on health care in year x and ( ) I x represents income in year x. Find a function H that represents total health care expenditures in year x. 116. Income Tax Suppose that ( ) I x represents the income of an individual in year x before taxes and ( ) T x represents the individual’s tax bill in year x. Find a function N that represents the individual’s net income (income after taxes) in year x. 117. Profit Function Suppose that the revenue R, in dollars, from selling x smartphones, in hundreds, is ( ) = − + R x x x 1.2 220 2 The cost C, in dollars, of selling x smartphones, in hundreds, is ( ) = − + + C x x x x 0.05 2 65 500. 3 2 (a) Find the profit function, ( ) ( ) ( ) = − P x R x C x . (b) Find the profit if = x 15 hundred smartphones are sold. (c) Interpret ( ) P 15 . 118. Population as a Function of Age The function ( ) = = − + P P a a a 0.027 6.530 363.804 2 represents the population P (in millions) of people in the United States who were at least a years old. (a) Identify the dependent and independent variables. (b) Evaluate ( ) P 20 . Explain the meaning of ( ) P 20 . (c) Evaluate ( ) P 0 . Explain the meaning of ( ) P 0 . Source: U.S. Census Bureau. 119. Stopping Distance When the driver of a vehicle observes an impediment, the total stopping distance involves both the reaction distance R (the distance the vehicle travels while the foot of the driver is moved to the brake pedal) and the braking distance B (the distance the vehicle travels once the brakes are applied). For a car traveling at a speed of v miles per hour, the reaction distance R, in feet, can be estimated by ( ) = R v v 2.2 . Suppose that the braking distance B, in feet, for a car is given by ( ) = + − B v v v 0.05 0.4 15. 2 (a) Find the stopping distance function ( ) ( ) ( ) = + D v R v B v (b) Find the stopping distance if the car is traveling at a speed of 60 mph. (c) Interpret ( ) D 60 . 120. Some functions f have the property that ( ) ( ) ( ) + = + f a b f a f b for all real numbers a and b.Which of the following functions have this property? (a) ( ) = h x x2 (b) ( ) = g x x2 (c) ( ) = − F x x5 2 (d) ( ) = G x x 1 121. Challenge Problem Find the difference quotient of the function ( ) = f x x. 3 (Hint: Factor using ( ) ( ) − = − + + a b a b a ab b 3 3 2 2 with = + a x h 3 and = b x. 3 ) 122. Challenge Problem If ( ) + − = − f x x x 4 5 4 3 2, 2 find ( ) f 1 . 123. Challenge Problem Find the domain of ( ) = + − − f x x x 1 7 3 1 . 2 110. Effect of Gravity on Jupiter If a rock falls from a height of 20 meters on the planet Jupiter, its height H (in meters) after x seconds is approximately ( ) = − H x x 20 13 2 (a) What is the height of the rock when = x 1 second? When = x 1.1 seconds? When = x 1.2 seconds? (b) When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters? (c) When does the rock strike the ground? 111. Cost of Transatlantic Travel A Boeing 787 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by ( ) = + + C x x x 100 10 36,000 where x is the ground speed ( ) ± airspeed wind . (a) What is the cost per passenger for quiescent (no wind) conditions? (b) What is the cost per passenger with a head wind of 50 miles per hour? (c) What is the cost per passenger with a tail wind of 100 miles per hour? (d) What is the cost per passenger with a head wind of 100 miles per hour? 112. Cross-sectional Area The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function ( ) = − A x x x 4 1 , 2 where x represents the length, in feet, of half the base of the beam. See the figure. Determine the cross-sectional area of the beam if the length of half the base of the beam is as follows: (a) One-third of a foot (b) One-half of a foot (c) Two-thirds of a foot 113. Economics The participation rate is the number of people in the labor force divided by the civilian population (excludes military). Let ( ) L x represent the size of the labor force in year x, and ( ) P x represent the civilian population in year x. Determine a function that represents the participation rate R as a function of x. 114. Crimes Suppose that ( ) V x represents the number of violent crimes committed in year x and ( ) P x represents the number of property crimes committed in year x. Determine a function T that represents the combined total of violent crimes and property crimes in year x. A(x) 5 4x 1 2 x2 x 1 Credit:World History Archive/Alamy Stock Photo
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