Question

The dimensions of a cylinder are changing, but the height is always equal to the diameter of the base of the cylinder. If the height is increasing at a speed of 3 inches per second, determine the speed at which the volume, V , is increasing (in cubic inches per second) when the height is 2 inches.

Answer 1

Answer:

dV/dt  = 9 cubic inches per second

Explanation:

Let the height of the cylinder is h

Diameter of cylinder = height of the cylinder = h

Radius of cylinder, r = h/2

dh/dt = 3 inches /s

Volume of cylinder is given by

$V = \pi r^{2}h$

put r = h/2 so,

$V = \pi \frac{h^{3}}{4}$

Differentiate both sides with respect to t.

$\frac{dV}{dt}=\frac{3h^{2}}{4}\times \frac{dh}{dt}$

Substitute the values, h = 2 inches, dh/dt = 3 inches / s

$\frac{dV}{dt}=\frac{3\times 2\times 2}{4}\times 3$

dV/dt  = 9 cubic inches per second

Thus, the volume of cylinder increases by the rate of 9 cubic inches per second.