hw8 - Granillo Yvette Homework 8 Due 3:00 am Inst Edward...

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Granillo, Yvette – Homework 8 – Due: Oct 20 2005, 3:00 am – Inst: Edward Odell 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 6 ft/sec, at what speed is the area of the ripple increasing when its radius is 2 feet? 1. speed = 22 π sq. ft/sec 2. speed = 22 sq. ft/sec 3. speed = 24 π sq. ft/sec correct 4. speed = 20 sq. ft/sec 5. speed = 23 sq. ft/sec 6. speed = 24 sq. ft/sec 7. speed = 20 π sq. ft/sec 8. speed = 21 π sq. ft/sec Explanation: The area, A , of a circle having radius r is given by A = πr 2 . Differentiating implicitly with respect to t we thus see that dA dt = 2 πr dr dt . When r = 2 , dr dt = 6 , therefore, the speed at which the area of the ripple is increasing is given by speed = 24 π sq. ft/sec . keywords: Stewart5e, 002 (part 1 of 1) 10 points A point is moving on the graph of xy = 3. When the point is at (2 , 3 2 ), its x -coordinate is increasing at a rate of 6 units per second. What is the speed of the y -coordinate at that moment and in which direction is it moving? 1. speed = 9 2 units/sec, decreasing y correct 2. speed = 11 2 units/sec, increasing y 3. speed = 13 2 units/sec, increasing y 4. speed = - 11 2 units/sec, decreasing y 5. speed = - 9 2 units/sec, decreasing y Explanation: Provided x, y 6 = 0, the equation xy = 3 can be written as y = 3 /x . Differentiating implicitly with respect to t we thus see that dy dt = - 3 x 2 dx dt . whenever x 6 = 0. When x = 2 , dx dt = 6 , therefore, the corresponding rate of change of the y -coordinate is given by dy dt fl fl fl x =2 = - 6 3 x 2 ·fl fl fl x =2 = - 9 2 . Consequently, the speed of the y -coordintate is 9 2 units per second and the negative sign indicates that the point is moving in the di- rection of decreasing y . keywords: Stewart5e, 003 (part 1 of 1) 10 points
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Granillo, Yvette – Homework 8 – Due: Oct 20 2005, 3:00 am – Inst: Edward Odell 2 A street light is on top of a 10 foot pole. A person who is 4 feet tall walks away from the pole at a rate of 3 feet per second. At what speed is the tip of the person’s shadow moving when he is 10 feet from the pole? 1. tip speed = 31 6 ft/sec 2. tip speed = 29 6 ft/sec 3. tip speed = 5 ft/sec correct 4. tip speed = 14 3 ft/sec 5. tip speed = 16 3 ft/sec Explanation: If x denotes the distance of the tip of the person’s shadow from the pole and y denotes the distance of the person from the pole, then the shadow and the lightpole are related in the following diagram (0 , 10) (10 , 4) y x By similar triangles, 4 x - y = 10 x , so (10 - 4) x = 10 y . Thus, after implicit differentiation with respect to t , (10 - 4) dx dt = 10 dy dt . When y = 10 , dy dt = 3 , therefore, tip speed = 5 ft/sec . keywords: Stewart5e, related rates, street light problem, implicit differentiation appli- cation 004 (part 1 of 1) 10 points At noon, ship A is 120 miles due west of ship B . Ship A is sailing south at 30 mph while ship B is sailing north at 10 mph.
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