HW09.pdf - young(toy68 HW09 villafuerte altu(53785 This...

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young (toy68) – HW09 – villafuerte altu – (53785) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points 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 2 feet? 1. speed = 18 sq. ft/sec 2. speed = 15 π sq. ft/sec 3. speed = 18 π sq. ft/sec 4. speed = 16 sq. ft/sec 5. speed = 15 sq. ft/sec 6. speed = 16 π sq. ft/sec correct 7. speed = 19 sq. ft/sec 8. speed = 17 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 = 4 , therefore, the speed at which the area of the ripple is increasing is given by speed = 16 π sq. ft/sec . 002(part1of2)10.0points A point is moving on the graph of 4 x 3 + 5 y 3 = xy . When the point is at P = parenleftBig 1 9 , 1 9 parenrightBig , its x -coordinate is decreasing at a speed of 7 units per second. What is the speed of the y -coordinate at that time? 1. speed y -coord = - 9 2 units/sec 2. speed y -coord = - 7 2 units/sec 3. speed y -coord = 9 2 units/sec 4. speed y -coord = 7 2 units/sec correct 5. speed y -coord = 5 2 units/sec Explanation: Differentiating 4 x 3 + 5 y 3 = xy implicitly with respect to t we see that 12 x 2 dx dt + 15 y 2 dy dt = y dx dt + x dy dt . Thus dy dt = parenleftbigg 12 x 2 - y x - 15 y 2 parenrightbigg dx dt . Now at P parenleftbigg 1 9 , 1 9 parenrightbigg , 12 x 2 - y = parenleftbigg 12 (9) 2 - 1 9 parenrightbigg = 1 (9) 2 (3) , while x - 15 y 2 = parenleftbigg 1 9 - 15 (9) 2 parenrightbigg = - 1 (9) 2 (6) . Hence, at P , dy dt = - 1 2 dx dt .
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young (toy68) – HW09 – villafuerte altu – (53785) 2 When the x -coordinate at P is decreasing at a rate of 7 units per second, therefore, dy dt = 7 parenleftbigg 1 2 parenrightbigg , so the speed y -coord = 7 2 units/sec . 003(part2of2)10.0points In which direction is the y -coordinate mov- ing at that time? 1. direction decreasing y 2. direction increasing y correct Explanation: Since dy dt = 7 2 , at P , the y -coordinate of the point is moving in the direction increasing y at that time. 004 10.0points If a snowball melts so that its surface area decreases at a rate of 1 cm 2 / min, find the rate at which the diameter decreases when the diameter is 28 cm. 1. 1 14 π cm / min 2. 1 56 π cm / min correct 3. 1 28 π cm / min 4. 1 112 π cm / min 5. 1 84 π cm / min Explanation: If the radius is r and the diameter is x = 2 r then r = 1 2 x and S = 4 πr 2 = 4 π parenleftbigg 1 2 x parenrightbigg 2 = πx 2 dS dt = dS dx dx dt = 2 πx dx dt dS dt = - 1 2 πx dx dt = - 1 dx dt = - 1 2 πx When x = 28 , dx dt = - 1 56 π , so the rate of decrease is 1 56 π cm / min. 005 10.0points Gravel is being dumped from a conveyor belt at a rate of 25 ft 3 / min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal.
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