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Unformatted text preview: white (taw933) HW08 ben-zvi (55600) 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 10.0 points A 15 foot ladder is leaning against a wall. If the foot of the ladder is sliding away from the wall at a rate of 12 ft/sec, at what speed is the top of the ladder falling when the foot of the ladder is 12 feet away from the base of the wall? 1. speed = 15 ft/sec 2. speed = 16 ft/sec correct 3. speed = 47 3 ft/sec 4. speed = 44 3 ft/sec 5. speed = 46 3 ft/sec Explanation: Let y be the height of the ladder when the foot of the ladder is x feet from the base of the wall as shown in figure x ft. 15 ft. We have to express dy/dt in terms of x, y and dx/dt . But by Pythagoras theorem, x 2 + y 2 = 225 , so by implicit differentiation, 2 x dx dt + 2 y dy dt = 0 . In this case dy dt =- x y dx dt . But again by Pythagoras, if x = 12, then y = 9. Thus, if the foot of the ladder is moving away from the wall at a speed of dx dt = 12 ft/sec , and x = 12, then the velocity of the top of the ladder is given by dy dt =- 4 3 dx dt . Consequently, the speed at which the top of the ladder is falling is speed = dy dt = 16 ft/sec . keywords: speed, ladder, related rates 002 10.0 points A circle of radius r has area A and circum- ference C are given respectively by A = r 2 , C = 2 r . If r varies with time t , for what value of r is the rate of change of A with respect to t equal to the rate of change of C with respect to t ? 1. r = 1 correct 2. r = 2 3. r = 2 4. r = 1 2 5. r = white (taw933) HW08 ben-zvi (55600) 2 6. r = 2 Explanation: Differentiating A = r 2 , C = 2 r implicitly with respect to t we see that dA dt = 2 r dr dt , dC dt = 2 dr dt . Thus the rate of change, dA/dt , of area is equal to the rate of change, dC/dt , of circum- ference when dA dt = dC dt , i.e. , when 2 r dr dt = 2 dr dt . This happens when r = 1 . 003 10.0 points Determine the value of dy/dt at x = 3 when y = x 2- 2 x and dx/dt = 3. 1. dy dt x =3 = 16 2. dy dt x =3 = 20 3. dy dt x =3 = 14 4. dy dt x =3 = 18 5. dy dt x =3 = 12 correct Explanation: Differentiating implicitly with respect to t we see that dy dt = (2 x- 2) dx dt = 3 (2 x- 2) . At x = 3, therefore, dy dt = 3(4) = 12 . 004 10.0 points A point is moving on the graph of xy = 2. When the point is at (3 , 2 3 ), 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 mov- ing? 1. speed =- 10 3 units/sec, increasing y 2. speed = 4 3 units/sec, decreasing y correct 3. speed = 7 3 units/sec, increasing y 4. speed =- 7 3 units/sec, decreasing y 5. speed =- 4 3 units/sec, decreasing y 6. speed = 10 3 units/sec, increasing y Explanation: Provided x, y = 0, the equation xy = 2 can be written as y = 2 /x . Di ff erentiating implicitly with respect to t we thus see that dy dt =- 2 x 2 dx dt ....
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- Spring '08
- Differential Calculus