Answer:
Step-by-step explanation:
This is a classic related rates problem. Gotta love calculus!
Start out with the formula for the volume of a cone, which is
and with what we know, which is
and the fact that the diameter = height (we will come back to that in a bit).
We need to find when h = 13
The thing we need to notice now is that there is no information given to us that involves the radius. It does, however, give us a height. We need to replace the r with something in terms of h. Let's work on that first.
We know that d = h. Because d = 2r, we can say that 2r = h, and solving for r gives us that .
Now we can rewrite the formula with that replacement:
Simplify that all the way down to
The derivative of that function with respect to time is
Filling in what we have gives us this:
Solve that for the rate of change of the height:
or in decimal form: