Related Rates: Conical Pile Height Problem | Calculus Question

[Jordan_]

I've been working on this problem for a while now and I can't seem to make it work. Maybe I could get a hint?

A conveyer belt system at a gravel pit pours washed sand onto the ground at a rate of 180m^3/h. The sand forms a conical pile with the height always one fifth the diameter of the base. Find out how fast the heith of the pile is increasing at the instant when the radius of the base is 6m.

I understand I'm given dv/dt = 180 as well as r = 6 and h = 2r/5 = 36/5

The equation I'm using is v = Bh/3 or v = (pi)(r^2)(h)/3

I go on to take the derivative and input my variables but I'm always missing dr/dt. Maybe a hint?

By the by the answer is stated as 1.59m/h

 

[StatusX]

You want an equation relating the volume of the pile to its height, which you can get by writing the radius in terms of the height. Then take the derivative of both sides.

 

[Jordan_]

Okay so if my volume equation is

v = pi(r^2)(h) and I want to get rid of the radius, I simple take the information I was given...that the heigh is one fifth the diameter of the circle and for the equation h = 2r/5. I then rearrange it to say r = 5h/2 and substitute that in?

Then it makes v = pi(5h/2)^2(h) which then gives me v = pi(25h^3)/4

when I take the derivative of that, and sub in the height which I get as 2.4,
I end up with dh/dt = 1.27m which isn't the right answer. What am I doing wrong?

 

[StatusX]

Well, the volume should be \pi r^2 h/3, but that mistake would give you an answer that's off by a factor of three, so you must be making another error in some calculation you haven't shown.

 

[Jordan_]

Sorry about that mate. Least I could have done is checked over my calculations before asking. I just did it again your way and got the answer. Thanks for the help bro it's appreciated.