Question

The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing when the diameter is 100 mm?

Answer 1

Answer:

Volume increasing rate = $62831.85 mm^3/s$

Explanation:

We rate of change of radius of sphere, $\frac{dr}{dt} =2mm/s$

Diameter of sphere = 100 mm

Radius of sphere = 50 mm

Volume of sphere, V = $\frac{4}{3} \pi r^3$

Rate of change of volume = $\frac{dV}{dt}$

$\frac{dV}{dt}=\frac{d}{dt} (\frac{4}{3} \pi r^3)=\frac{4}{3} \pi\frac{d}{dt}(r^3)=\frac{4}{3} \pi*3r^2*\frac{dr}{dt}$

Substituting known values

     $\frac{dV}{dt}= \frac{4}{3}* \pi*3*50^2*2=62831.85 mm^3/s$

Volume increasing rate = $62831.85 mm^3/s$