SOLUTION: GENERAL: Hailstones A hailstone (a small sphere of ice) is forming in the clouds so that its radius is growing at the rate of 1 millimeter per minute. How fast is its volume growin

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: GENERAL: Hailstones A hailstone (a small sphere of ice) is forming in the clouds so that its radius is growing at the rate of 1 millimeter per minute. How fast is its volume growin      Log On


   

Question 1125312: GENERAL: Hailstones A hailstone (a small sphere of ice) is forming in the clouds so that its radius is growing at the rate of 1 millimeter per minute. How fast is its volume growing at the moment when the radius is 2 millimeters? [Hint: The volume of a sphere of radius r is V = 4/3*pi r^3.]
Found 3 solutions by Alan3354, ikleyn, rothauserc:
Answer by Alan3354(69443) About Me  (Show Source):
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Hailstones A hailstone (a small sphere of ice) is forming in the clouds so that its radius is growing at the rate of 1 millimeter per minute. How fast is its volume growing at the moment when the radius is 2 millimeters?
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V+=+4pi%2Ar%5E3%2F3
dV/dt = 4pi*r^2*dr/dt
dV/dt = 4pi*4*1 = 16pi cubic mms/minute
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Hint: We don't need hints.

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
You are given  V(t) = V(r(t)) = %284%2F3%29%2Api%2Ar%28t%29%5E3,  where the radius  r  varies  such that  %28dr%29%2F%28dt%29 = 1 millimeter per minute.


Then  %28dV%29%2F%28dt%29 = 4%2Api%2Ar%5E2.%28dr%29%2Fdt%29 = 4%2A3.14%2A2%5E2.1 = 4*3.14*4 = 16pi = 50.24 mm^3 per minute.


Thus the instantaneous volume growing rate is  50.24 mm^3/minute.



Interesting, that if you calculate the averaged volume growing rate between  t= 2 seconds  and  t= 3 seconds, you will get another value


%28V%283%29+-+V%282%29%29%2F%283-2%29 = %284%2F3%29%2Api%2A3%5E3 - %284%2F3%29%2Api%2A2%5E3 = %284%2F3%29%2Api%2A%2827-8%29 = %2876%2F3%29%2Api = %2876%2F3%29%2A3.14 = 79.55  mm^3/minute.


But it only demonstrates the difference between these two conceptions: instantaneous and averaged rates.


Answer.   The instantaneous volume growing rate at this moment is  50.24 mm^3 per minute.



Answer by rothauserc(4718) About Me  (Show Source):
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take first derivative of V and r with respect to time
:
dV/dt = 3 * (4/3) * pi * r^2 * dr/dt = 4 * pi * r^2 * dr/dt
:
dV/dt = 4 * pi * 4 * 1 = 16 * pi is approximately 50.27 mm^3 per minute
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