# Related Rates Sand Pile Problem

1. Jan 11, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
A machine starts dumping sand at the rate of 20 m3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the base diameter. After 5 minutes, how fast is the height increasing? After 5 minutes, how fast is the area of the base increasing?

2. Relevant equations

V = (1/3)(pi)r^2(h)

3. The attempt at a solution

For the first question,
$h = 4r$
$\displaystyle V(t) = 20t = \frac{1}{3}\pi r^2 h = \frac{\pi}{48}h^3$
From this equation I get
$\displaystyle h = (\frac{960}{\pi}t)^{\frac{1}{3}}$
then
$\frac{dh}{dt} = \frac{320}{\pi ((\frac{960}{\pi})t)^{\frac{2}{3}}}$
When 5 is substituted for t, I get 0.77 m^3/min. Is this correct?

Also, I am not sure how to approach the second problem. I know that the area of the base is $\pi r^2$, but I am not sure how to proceed...

Last edited: Jan 11, 2016
2. Jan 11, 2016

### Thewindyfan

Use the fact that height is equal to twice the diameter of the base at all times.

Last edited: Jan 11, 2016
3. Jan 11, 2016

### Mr Davis 97

I did that for the first problem. I don't see how it will work for the second problem though.

4. Jan 11, 2016

### Thewindyfan

EDIT: ah never mind, fairly straightforward problem. Haven't seen one like this in awhile though.

Method 1: If you're know what dH/dt is, try applying that to see what the radius would be.

EDIT: you did the first part right.
Suggested Hint: Try rewriting the area equation.

...and check your units to part 1.

Last edited: Jan 11, 2016