Question

Suppose the radius of the sphere is increasing at a constant rate of 0.3 centimeters per second. At the moment when the radius is 24 centimeters, the volume is increasing at a rate of?

Answer 1

At the moment when the radius is 24 centimeters, the volume is increasing at a rate of 2171.47 cm³/min.

Step-by-step explanation:

We have equation for volume of a sphere

             $V=\frac{4}{3}\pi r^3$

where r is the radius

Differentiating with respect to time,

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

Given that

           Radius, r = 24 cm

           $\frac{dr}{dt}=0.3cm/s$

Substituting

           $\frac{dV}{dt}=4\pi r^2\times \frac{dr}{dt}\\\\\frac{dV}{dt}=4\pi \times 24^2\times 0.3\\\\\frac{dV}{dt}=2171.47cm^3/min$

At the moment when the radius is 24 centimeters, the volume is increasing at a rate of 2171.47 cm³/min.