SECTION 3.3 Quadratic Functions and Their Properties 169 92. Minimizing Marginal Cost (See Problem 91.) The marginal cost C (in dollars) of manufacturing x smartphones (in thousands) is given by C x x x 5 200 4000 2 ( ) = − + (a) How many smartphones should be manufactured to minimize the marginal cost? (b) What is the minimum marginal cost? 93. Business The monthly revenue R achieved by selling x wristwatches is R x x x 75 0.2 .2 ( ) = − The monthly cost C of selling x wristwatches is C x x 32 1750 ( ) = + (a) How many wristwatches must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as P x R x C x . ( ) ( ) ( ) = − What is the profit function? (c) How many wristwatches must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue. 94. Business The daily revenue R achieved by selling x boxes of candy is R x x x 9.5 0.04 .2 ( ) = − The daily cost C of selling x boxes of candy is C x x 1.25 250. ( ) = + (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as P x R x C x . ( ) ( ) ( ) = − What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue. 95. Stopping Distance An accepted relationship between stopping distance d (in feet), and the speed v of a car (in mph), is d v v 1.1 0.06 2 = + on dry, level concrete. (a) How many feet will it take a car traveling 45 mph to stop on dry, level concrete? (b) If an accident occurs 200 feet ahead of you, what is the maximum speed you can be traveling to avoid being involved? 96. Playing Field Although a stadium field appears to be flat, its surface is actually shaped like a parabola so that rainwater will run off to the sides. If we take a cross section of a particular field along its width, the surface can be modeled by = − + y x x 0.000092 0.020741 , 2 where x is the distance in feet from the left end of the field and y is the height, in feet, of the field. (a) What is the width of the field? (b) What is the crown, or maximum height in the middle of the field? Crown Cross section of a field with central crown. 97. Chemical Reactions A self-catalytic chemical reaction results in the formation of a compound that causes the formation ratio to increase. If the reaction rate V is modeled by V x kx a x x a , 0 ( ) ( ) = − ≤ ≤ where k is a positive constant, a is the initial amount of the compound, and x is the variable amount of the compound, for what value of x is the reaction rate a maximum? 87. Analyzing the Motion of a Projectile A projectile is fired from a cliff 200 feet above the water at an inclination of 45° to the horizontal, with a muzzle velocity of 50 feet per second. The height h of the projectile above the water is modeled by h x x x 32 50 200 2 2 ( ) = − + + where x is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function h, x 0 200. ≤ ≤ (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff? 88. Analyzing the Motion of a Projectile A projectile is fired at an inclination of 45° to the horizontal, with a muzzle velocity of 100 feet per second. The height h of the projectile is modeled by h x x x 32 100 2 2 ( ) = − + where x is the horizontal distance of the projectile from the firing point. (a) At what horizontal distance from the firing point is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the firing point will the projectile strike the ground? (d) Graph the function h, x 0 350. ≤ ≤ (e) Use a graphing utility to verify the results obtained in parts (b) and (c). (f) When the height of the projectile is 50 feet above the ground, how far has it traveled horizontally? 89. Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that when the unit price is p dollars, the revenue R (in dollars) is R p p p 4 4000 2 ( ) = − + What unit price p maximizes revenue? What is the maximum revenue? 90. Maximizing Revenue A lawn mower manufacturer has found that the revenue, in dollars, from sales of zero-turn mowers is a function of the unit price p, in dollars, that it charges. If the revenue R is R p p p 1 2 2900 2 ( ) = − + what unit price p should be charged to maximize revenue? What is the maximum revenue? 91. Minimizing Marginal Cost The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is $6.20, it costs $6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C (in dollars) to produce x thousand digital music players is given by the function C x x x 140 7400 2 ( ) = − + (a) How many players should be produced to minimize the marginal cost? (b) What is the minimum marginal cost?
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