the radius of a sphere is expanding at a rate of inches per minute. determine the rate at which the volume is changing with respect to time when?
Assume that the radius r of a sphere is expanding at a rate of 7 in. /min. The volume of a sphere is V=43πr3.
Determine the rate at which the volume is changing with respect to time when r=16 in.
I know I need to find the derivative of volume, and I think solve for dr/dV and then plug in when r=16. I’ve tried and I keep getting all kinds of wrong answers. Help would be greatly appreciated.
the radius of a sphere is expanding at a rate of inches per minute. determine the rate at which the volume is changing with respect to time when?
The volume of the sphere is:
V=4π3r3
Differentiating volume with respect to radius gives:
dVdr=4πr2
However, we want the differential of volume with respect to time. To do this we can multiply both sides by drdt:
drdtdVdr=4πr2drdt
Cancelling out the dr on the lefthand side gives us the differential of volume with respect to time:
dVdt=4πr2drdt
Now, plugging in the rate of 7in.min. for drdt and 16in. for the radius, the rate at which the volume is changing is:
dVdt=4π(16in.)2(7in.min.)=4π(256in.2)(7in.min.)=
7168π in.3min.