Find the rate of change of the volume for each of the radii given below if dr/dt is 5 inches per min and h=12r

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 5 inches per minute and h = 12r.

V = (1/3)𝜋r²h

(a) r = 8 in

V'= ? (𝜋 in³/min)

(b) r = 15 in

V'= ? (𝜋 in³/min)

Thank you so much in advance for any guidance. As I am not sure how to approach this

Answer 1

Since you are told that h = 12r, you can write the volume equation V = (1/3)𝜋r²h as:

V = (1/3)𝜋r²(12r) by substituting "12r" for "h"

Simplifying, it becomes:

V = 4πr³ which is volume as a function of radius or:

V(r) = 4πr³

Now, to find the rate of change of the volume (or dV/dt), you just need to take the derivative with respect to "t". Since the function is written as V(r), to take dV/dt requires the use of the chain rule.

dV/dt = dV/dr • dr/dt

Taking dV/dr we get 12πr² so dV/dt = 12πr²• dr/dt

Plugging in the values we have (r = 8 and dr/dt = 5) we get dV/dt = 12π(8²)(5) = 3840π in³/min

To do problem b), just plug in r = 15 so dV/dt = 12π(15²)(5)