Question

The radius of a cylinder is increasing at the rate 2 cm/sec. and its altitude is decreasing at the rate of 3 cm/sec. Find the rate of change of volume when radius is 3 cm and altitude 5 cm.

Answer 1

$$\text{ Let r e the radius,h be the height and V be the volume of the cylinder at any time t. Then },$$
$$V=\pi r^{2}h$$
$$\Rightarrow \frac{dV}{dt}=2\pi rh\frac{dr}{dt}+\pi r^2\frac{dh}{dt}$$
$$\Rightarrow \frac{dV}{dt}=\pi r(2h\frac{dr}{dt}+r\frac{dh}{dt})$$
$$\Rightarrow \frac{dV}{dt}=\pi \times 3(2\times 5\times 2+3\times -3)$$
$$\Rightarrow \frac{dV}{dt}=3\pi (20-9)$$
$$\Rightarrow \frac{dV}{dt}=33\pi {\text{ cm}}^3/\sec$$