HW08-solutions-1

HW08-solutions-1 - white(taw933 HW08 ben-zvi(55600 This...

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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 – fnd 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 fgure 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 di±erentiation, 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 = v v v dy dt v v v = 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 = πr 2 , C = 2 πr . 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 π 4. r = 1 2 5. r = π
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white (taw933) – HW08 – ben-zvi – (55600) 2 6. r = π 2 Explanation: Diferentiating A = πr 2 , C = 2 implicitly with respect to t we see that dA dt = 2 dr dt , dC dt = 2 π 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 dr dt = 2 π 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 v v v x =3 = 16 2. dy dt v v v x =3 = 20 3. dy dt v v v x =3 = 14 4. dy dt v v v x =3 = 18 5. dy dt v v v x =3 = 12 correct Explanation: Diferentiating 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 n = 0, the equation xy = 2 can be written as y = 2 /x . Diferentiating implicitly with respect to t we thus see that dy dt = 2 x 2 dx dt .
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HW08-solutions-1 - white(taw933 HW08 ben-zvi(55600 This...

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