The radius of a sphere is r cm at time t seconds. Find the radius when the rate of increase of the surface area is equal to the rate of increase of the radius.

Ok, the surface area of a sphere (SA) is 4*pi*r^2.

SA = 4*pi*r^2

We want to take the derivative of this equation with respect to time:

dSA/dt = 4*pi*2*r*(dr/dt) = 8*pi*r*(dr/dt)

We know from the problem that the rate of increase of the surface area is equal to the rate of increase of the radius. That is, dSA/dt = dr/dt. So, I'm going to replace dSA/dt with dr/dt:

(dSA/dt) = 8*pi*r*(dr/dt)

(dr/dt) = 8*pi*r*(dr/dt)

Divide through by dr/dt:

1 = 8*pi*r

Solve for r:

r = 1/(8*pi)

Nov 29th, 2014

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