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)πr2ha) r = 5 in (b) r = 17 in
Accepted Solution
A:
Answer:a) [tex]5400\pi\text{ cubic inches per min}[/tex]b) [tex]62424\pi\text{ cubic inches per min}[/tex]Step-by-step explanation:Since, volume of a cone is,[tex]V =\frac{1}{3}\pi r^2 h-----(1)[/tex]Where,r = radius,h = height,Here, h = 24r,From equation (1),[tex]V =\frac{1}{3}\pi r^2 (24r)[/tex][tex]=8\pi r^3[/tex]Differentiating with respect to t(time),[tex]\frac{dV}{dt}=24\pi r^2\frac{dr}{dt}[/tex]We have,[tex]\frac{dr}{dt}=9\text{ inches per minute}[/tex][tex]\implies \frac{dV}{dt}=216\pi r^2[/tex]a) r = 5 in,The rate of change of the volume,[tex]\frac{dV}{dt}=216\pi (5)^2 = 216\pi(25) = 5400\pi\text{ cubic inches per min}[/tex]b) r = 17 in,The rate of change of volume,[tex]\frac{dV}{dt}=216\pi (17)^2 = 216\pi(289) = 62424\pi\text{ cubic inches per min}[/tex]