Homework Help: calculus

Posted by Yoona on Tuesday, January 3, 2012 at 12:02am.

3. The radius r of a sphere is increasing at a constant rate of 0.04 centimeters per second. (Note: The volume of a sphere with radius r is v=4/3pir^3 ).
a. At the time when the radius of the sphere is 10 cm, what is the rate of increase of its volume?
b. At the time when the volume of the sphere is 36pi cubic centimeters, what is the rate of increase of the area of a cross section through the center of the sphere?
c. At the time when the volume and the radius of the sphere are increasing at the same numerical rate, what is the radius?

calculus - Reiny, Tuesday, January 3, 2012 at 9:44am
dV/dt = 4πr^2 dr/dt

a) when r=10 and dr/dt = .04
dV/dt = 4π(100)(.04) = ...

b) when V = 36π
36π = (4/3)πr^3
27 = r^3
r = 3
cross section is a circle, A = πr^2
dA/dt = 2πr dr/dt
= 2π(3)(.04)
= ..

c) dV/dt = 4πr^2 dr/dt, but dr/dt = dV/dt = .04
.04 = 4πr^2 (.04)
4πr^2 = 1
r^2 = 1/(4π)
r = 1/(2√π) = ...

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