SOLUTION: a conical tank with vertex at the bottom is being filled at a rate of 4 cubic m per minute. the tank is 24 m deep and 12 m at the top. find the rate of change of the depth of the w

Algebra ->  Volume -> SOLUTION: a conical tank with vertex at the bottom is being filled at a rate of 4 cubic m per minute. the tank is 24 m deep and 12 m at the top. find the rate of change of the depth of the w      Log On


   

Question 1102126: a conical tank with vertex at the bottom is being filled at a rate of 4 cubic m per minute. the tank is 24 m deep and 12 m at the top. find the rate of change of the depth of the waste when the water id 8m deep
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


You didn't specify whether the 12m is the radius or the diameter of the top of the tank. By the way you state the problem, I assume it is the full width (diameter) of the top of the tank.

Then the radius of the top of the tank is 6m; since the depth of the tank is 24m, the radius is 1/4 of the depth. As the tank is being filled, the volume of water forms a cone that is similar to the whole tank; so at any time the radius of the surface of the water in the tank is 1/4 the depth of the water.

So the volume of water when the depth is h is


dV%2Fdt+=+%281%2F16%29%28pi%29h%5E2%28dh%2Fdt%29

dh%2Fdt+=+%2816%2F%28%28pi%29h%5E2%29%29%2A%28dV%2Fdt%29

We know dV/dt is 4; when the depth h is 4, dh/dt is
%28%2816%2F%28%28pi%294%5E2%29%29%2A4%29+=+4%2Fpi