{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW10-solutions - wei(jw35975 HW10 milburn(54685 This...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
wei (jw35975) – HW10 – milburn – (54685) 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 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 half the rate of change of C with respect to t ? 1. r = 1 2. r = 1 2 correct 3. r = π 4. r = π 2 5. r = 2 6. r = 2 π Explanation: Differentiating A = πr 2 , C = 2 πr implicitly with respect to t we see that dA dt = 2 πr dr dt , dC dt = 2 π dr dt . Thus the rate of change, dA/dt , of area is half the rate of change, dC/dt , of circumference when dA dt = 1 2 dC dt , i.e. , when 2 πr dr dt = 1 2 parenleftBig 2 π dr dt parenrightBig . This happens when r = 1 2 . 002 10.0points A point is moving on the graph of 3 x 3 + 5 y 3 = xy. When the point is at P = parenleftBig 1 8 , 1 8 parenrightBig , its y -coordinate is increasing at a speed of 7 units per second. What is the speed of the x -coordinate at that time and in which direction is the x - coordinate moving? 1. speed = 48 units/sec , decreasing x 2. speed = 51 units/sec , increasing x 3. speed = 48 units/sec , increasing x 4. speed = 49 units/sec , decreasing x correct 5. speed = 50 units/sec , increasing x 6. speed = 51 units/sec , decreasing x 7. speed = 49 units/sec , increasing x 8. speed = 50 units/sec , decreasing x Explanation: Differentiating 3 x 3 + 5 y 3 = xy implicitly with respect to t we see that 9 x 2 dx dt + 15 y 2 dy dt = y dx dt + x dy dt . Thus dx dt = parenleftBig x - 15 y 2 9 x 2 - y parenrightBig dy dt .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
wei (jw35975) – HW10 – milburn – (54685) 2 Now at P , x - 15 y 2 = - 7 64 , while 9 x 2 - y = 1 64 . Hence, at P , dx dt = - 7 dy dt . When the y -coordinate at P is increasing at a rate of 7 units per second, therefore, dx dt = - 49 . Consequently, the x -coordinate is moving at speed = 49 units/sec , and the negative sign indicates that it is mov- ing in the direction of decreasing x . 003 10.0points If the radius of a melting snowball decreases at a rate of 2 ins/min, find the rate at which the volume is decreasing when the snowball has diameter 2 inches. 1. rate = 10 π cu.ins/min 2. rate = 7 π cu.ins/min 3. rate = 8 π cu.ins/min correct 4. rate = 9 π cu.ins/min 5. rate = 11 π cu.ins/min Explanation: The volume, V , of a sphere of radius r is given by V = 4 3 πr 3 . Thus by implicit differentiation, dV dt = 4 πr 2 dr dt = - 8 πr 2 , since dr/dt = - 2 ins/min. When the snow- ball has diameter 2 inches, therefore, its ra- dius r = 1 and dV dt = - 8 (1) π . Consequently, when the snowball has diam- eter 2 inches, the volume of the snowball is decreasing at a rate = 8 π cu.ins/min . 004 10.0points A street light is on top of a 12 foot pole. Joe, who is 3 feet tall, walks away from the pole at a rate of 3 feet per second. At what speed is the tip of Joe’s shadow moving from the base of the pole when he is 12 feet from the pole?
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}