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M408K Hm. 8 Solutions

M408K Hm. 8 Solutions - Mahon Kevin Homework 8 Due 3:00 am...

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Mahon, Kevin – 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 4 ft/sec, at what speed is the area of the ripple increasing when its radius is 6 feet? 1. speed = 47 π sq. ft/sec 2. speed = 49 sq. ft/sec 3. speed = 45 sq. ft/sec 4. speed = 46 π sq. ft/sec 5. speed = 45 π sq. ft/sec 6. speed = 47 sq. ft/sec 7. speed = 48 sq. ft/sec 8. speed = 48 π sq. ft/sec correct 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 = 6 , dr dt = 4 , therefore, the speed at which the area of the ripple is increasing is given by speed = 48 π sq. ft/sec . keywords: Stewart5e, 002 (part 1 of 1) 10 points A point is moving on the graph of xy = 5. When the point is at (3 , 5 3 ), its x -coordinate is increasing at a rate of 1 units per second. What is the speed of the y -coordinate at that moment and in which direction is it moving? 1. speed = - 5 9 units/sec, decreasing y 2. speed = 14 9 units/sec, increasing y 3. speed = 5 9 units/sec, decreasing y correct 4. speed = - 14 9 units/sec, decreasing y 5. speed = 23 9 units/sec, increasing y Explanation: Provided x, y 6 = 0, the equation xy = 5 can be written as y = 5 /x . Differentiating implicitly with respect to t we thus see that dy dt = - 5 x 2 dx dt . whenever x 6 = 0. When x = 3 , dx dt = 1 , therefore, the corresponding rate of change of the y -coordinate is given by dy dt fl fl fl x =3 = - 5 x 2 ·fl fl fl x =3 = - 5 9 . Consequently, the speed of the y -coordintate is 5 9 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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Mahon, Kevin – 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 5 feet tall walks away from the pole at a rate of 4 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 = 42 5 ft/sec 2. tip speed = 39 5 ft/sec 3. tip speed = 41 5 ft/sec 4. tip speed = 8 ft/sec correct 5. tip speed = 38 5 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 , 5) y x By similar triangles, 5 x - y = 10 x , so (10 - 5) x = 10 y . Thus, after implicit differentiation with respect to t , (10 - 5) dx dt = 10 dy dt . When y = 10 , dy dt = 4 , therefore, tip speed = 8 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 150 miles due west of ship B . Ship A is sailing south at 10 mph while ship B is sailing north at 30 mph.
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