4. 0/3 points | Previous Answers My Note A ruptured oil tanker causes a circular

Question

4. 0/3 points | Previous Answers My Note A ruptured oil tanker causes a circular oil slick on the surface of the ocean. Pay close attention to the units in this problem. (a) When the radius of the oil slick is 100 meters, the radius is expanding by 0.4 meter/minute. At that moment, how fast is the area of the slick expanding? Convert your answer to square meters per second. Exact: | 80 m2/sec Rounded to 2 decimal places: 20 X m2/sec (b) Suppose we also know that the thickness of the oil slick remains uniform and the volume of oil spilled remains fixed. At the time referred to in part (a), the thickness of the oil slick is 0.05 meters. How fast is the thickness of the oil slick decreasing? Convert your answer to milimeters per second. (Recall: 1 m1000 mm) Exact: mm/sec

Explanation / Answer

4)area A=pi r2

differentiate with respect to time t

dA/dt =pi* 2r* dr/dt

r =100m,dr/dt =0.4m/min=0.4/60 m/s

dA/dt =pi* 2*100* 0.4/60 m2/s

dA/dt =(4/3)pi m2/s

area changing at (4/3)pi m2/s

area changing at 4.19 m2/s

b)volume v=pi r2 *p ,where p =thickness

v=pi *1002 *0.05

v=500pi

volume is constant

500pi=pi r2 *p

500= r2 *p

differentiate with respect to time

0=(2rp dr/dt +r2 dp/dt)

dp/dt=-(2rp dr/dt)r2

dp/dt=-(2(p/r) dr/dt)

dp/dt=-(2(0.05/100) 0.4/60) m/s

dp/dt=-0.0067 mm/s

thickness decreasing at 0.0067 mm/s

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