Question

The formula for the volume of a right cylinder is V = (pi)r^2h. Find the rate of change of the volume when r = 2 inches if dr/dt = 1/8 in./min and h = 5r. Show all your work.

Answer 1

Answer:

The volume is increasing at a rate of 15π/2 squared inches per minute.

Step-by-step explanation:

The formula for the volume of a right cylinder is:

$V=\pi r^2h$

We want to find dV/dt when r = 2, dr/dt = 1/8 in/min, and h = 5r.

Since h = 5r:

$V=\pi(r)^2(5r)=5\pi r^3$

Differentiate both sides with respect to t:

$\displaystyle \frac{dV}{dt}=15\pi r^2\frac{dr}{dt}$

Since r = 2 and dr/dt = 1/8:

$\displaystyle \frac{dV}{dt}=15\pi(2)^2\Big(\frac{1}{8}\Big)=\frac{15\pi}{2}$

So, the volume is increasing at a rate of 15π/2 squared inches per minute.