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Unformatted text preview: Version 117 – K Exam 2 – Hamrick – (54868) 1 This printout should have 19 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the rate at which the surface area of a cube is changing with respect to its side length x when x = 2 cm. 1. rate = 64 π cm 2 / cm 2. rate = 16 π cm 2 / cm 3. rate = 48 cm 2 / cm 4. rate = 24 cm 2 / cm correct 5. rate = 4 cm 2 / cm Explanation: For a cube with side length x its surface area = 6 x 2 . Now the rate at which the surface area is changing with respect to x is the derivative of surface area with respect to x . Thus rate = d dx (surface area) = 12 x . Consequently, when x = 2 cm., rate = 24 cm 2 / cm . 002 10.0 points The cost function for Levi Strauss to pro duce x pairs of blue jeans is C ( x ) = 3700 + 4 x − 2 25 x 2 + 9 10000 x 3 . Find the marginal cost to Levi Strauss of producing 100 pairs of blue jeans. 1. marginal cost = $15 per pair correct 2. marginal cost = $16 per pair 3. marginal cost = $13 per pair 4. marginal cost = $12 per pair 5. marginal cost = $14 per pair Explanation: By definition, the Marginal cost is the derivative, C ′ ( x ), of the cost function C ( x ). Now C ′ ( x ) = 4 − 4 25 x + 27 10000 x 2 . When x = 100, therefore, marginal cost = C ′ (100) = $15 per pair . 003 10.0 points A street light is on top of a 14 foot pole. A person who is 4 feet tall walks away from the pole at a rate of 5 feet per second. At what speed is the length of the person’s shadow growing? 1. speed = 2 ft/sec correct 2. speed = 19 10 ft/sec 3. speed = 9 5 ft/sec 4. speed = 17 10 ft/sec 5. speed = 8 5 ft/sec Explanation: If x denotes the length of the person’s shadow and y denotes the distance of the person from the pole, then shadow and the lightpole are related in the following diagram Version 117 – K Exam 2 – Hamrick – (54868) 2 14 4 y x x + y By similar triangles, 4 x = 14 x + y , so 4 y = (14 − 4) x . Thus, after implicit differ entiation with respect to t , 4 dy dt = (14 − 4) dx dt . When dy/dt = 5 , therefore, the length of the person’s shadow is growing with speed = 2 ft/sec . 004 10.0 points A weather balloon is rising vertically at 60 meters per minute. An observer is standing on the ground 54 meters from the point at which the balloon was released. Determine (in meters per minute) the rate at which the distance between the feet of the observer and the balloon is changing when the balloon is 72 meters high. ( Hint : remember 345 right triangles.) 1. rate = 50 meters/min 2. rate = 48 meters/min correct 3. rate = 49 meters/min 4. rate = 51 meters/min 5. rate = 52 meters/min Explanation: Let h be the height of the balloon t seconds after it is released and s the distance of the balloon from the observer as shown in the figure 54 balloon h s observer Then by Pythagoras’ theorem, h 2 + 54 2 = s 2 ....
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 Fall '08
 schultz
 Differential Calculus

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