[calculus] Tough problem... any help?

[orbitalfreak]

Howdy.

I'm an undergraduate math major working as a math tutor at my university, and had a student come in today with a related-rates problem. Another tutor and I both spend about 45 minutes on it, making almost zero progress. Any help would be appreciated.

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A conveyor belt carries sand and deposits it at a rate of 50 cubic metres per minute (100m^3/min). The sand piles up in a cone, whose height is 3/8 the diameter of the cone. Find the rate of change of (a) the height of the cone, and (b) the radius of the cone, when the cone is 4 metres high.

I'm probably missing something obvious, but cannot seem to work this problem.

 

[Hypersmurf]

The radius of the cone is easy.

h = 3d/8
h = ¾r
r = 4h/3
r = 16/3 m when h = 4m.

The rate of change of the height:

We know that, for a cone, V = 1/3 pi r²h.
We know that, for this cone, h = ¾r.
So, for this cone, V = 3pi/16 h³.

We also know that V = 100t m³ (t measured in minutes).

So, h = cube root (1600/3pi t), or cube root (1600/3pi) x t^(1/3).

So, the rate of change of h, dh/dt = 1/3 x cube root (1600/3pi) x t^(-2/3).

-Hyp.