As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as its radius. If the height of the pile is increasing at a rate of 6in/min, find the rate at...

As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as its radius. If the height of the pile is increasing at a rate of 6in/min, find the rate at which sand is leaking out when the altitude is 10 inches.

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

We can find the rate at which sand is leaking in volume per time. It is natural to measure time in minutes and volume in cubical inches for this problem.

Denote the radius of the cone (and its altitude) as `r(t)` (in inches). The rate of change of the height (and radius) is `r'(t)` and it is given to be `6` inches per minute. We also know that the volume of a (right circular) cone is `V = 1/3 \pi r^2 h,` where `h` is the altitude. In our case it gives  `V(t)=\pi/3 r^3(t).`

 

To find the rate of leaking, we need to take the derivative (use the Chain Rule):

`V'(t) = (\pi/3 r^3(t))' = \pi r^2(t)r'(t) = 6\pi r^2(t).`

 

For the moment when `r(t) = 10,` this derivative will be

`6\pi*10^2 = 600\pi approx 1885`  (cubic inches per minute).

 

When the altitude is 10 inches, the rate at which sand is leaking out is about 1885 cubic inches per minute.