400 CHAPTER 3 Polynomial and Rational Functions 116. Braking Distance The grade x of a hill is a measure of its steepness. For example, if a road rises 10 ft for every 100 ft of horizontal distance, then it has an uphill grade of x = 10 100 , or 10%. Grades are typically kept quite small—usually less than 10%. The braking distance D for a car traveling at 50 mph on a wet, uphill grade is given by the rational function D1x2 = 2500 3010.3 + x2 . (Data from Haefner, L., Introduction to Transportation Systems, Holt, Rinehart and Winston.) 10 ft 100 ft (a) Evaluate D10.052 to the nearest foot and interpret the result. (b) Describe what happens to braking distance as the hill becomes steeper. Does this agree with your driving experience? (c) Estimate, to the nearest tenth of a percent, the grade associated with a braking distance of 220 ft. 117. Tax Revenue Economist Arthur Laffer has been a center of controversy because of his Laffer curve, an idealized version of which is shown here. x y Maximum revenue Tax rate percent 0 x 1 x2 100 According to this curve, increasing a tax rate, say from x1 percent to x2 percent on the graph, can actually lead to a decrease in government revenue. All economists agree on the endpoints, 0 revenue at tax rates of both 0% and 100%, but there is much disagreement on the location of the rate x1 that produces maximum revenue. Suppose an economist studying the Laffer curve produces the rational function R1x2 = 80x - 8000 x - 110 , where R1x2 is government revenue in tens of millions of dollars for a tax rate of x percent, with the function valid for 55 … x … 100. Find the revenue for the following tax rates. Round to the nearest tenth if necessary. (a) 55% (b) 60% (c) 70% (d) 90% (e) 100% 118. Tax Revenue Suppose an economist determines the rational function R1x2 = 60x - 6000 x - 120 , where y = R1x2 is government revenue in tens of millions of dollars for a tax rate of x percent, with y = R1x2 valid for 50 … x … 100. Find the revenue for each tax rate. Round to the nearest tenth if necessary. (a) 50% (b) 60% (c) 80% (d) 100%
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