Question

The formula for the volume of a cone is given below. Find the rate of change of the volume for each of the radii given below if dr/dt is 9 inches per minute and h = 24r.


V = (1/3)πr2h


a) r = 5 in


(b) r = 17 in

Answer 1

Answer:

a) $5400\pi\text{ cubic inches per min}$

b) $62424\pi\text{ cubic inches per min}$

Step-by-step explanation:

Since, volume of a cone is,

$V =\frac{1}{3}\pi r^2 h-----(1)$

Where,

r = radius,

h = height,

Here, h = 24r,

From equation (1),

$V =\frac{1}{3}\pi r^2 (24r)$

$=8\pi r^3$

Differentiating with respect to t(time),

$\frac{dV}{dt}=24\pi r^2\frac{dr}{dt}$

We have,

$\frac{dr}{dt}=9\text{ inches per minute}$

$\implies \frac{dV}{dt}=216\pi r^2$

a) r = 5 in,

The  rate of change of the volume,

$\frac{dV}{dt}=216\pi (5)^2 = 216\pi(25) = 5400\pi\text{ cubic inches per min}$

b) r = 17 in,

The rate of change of volume,

$\frac{dV}{dt}=216\pi (17)^2 = 216\pi(289) = 62424\pi\text{ cubic inches per min}$