SOLUTION: Hello, please help me with this question. THANKYOU! Sand is poured into a conical pile with the height of the pile equalling the diameter of the pile. If the sand is poured at

Question 1103812: Hello, please help me with this question. THANKYOU!
Sand is poured into a conical pile with the height of the pile equalling the diameter of the pile. If
the sand is poured at a constant rate of 5m3 / s , at what rate is the height of the pile increasing
when the height is 2 meters?
Answer by greenestamps(13208) (Show Source): You can put this solution on YOUR website!

The volume of sand in the pile at any time is

We are given that dV/dt is 5 (m^3/sec); we need to find dh/dt at the moment the height h is 2.

Since we need to find dh/dt, we need to have the volume formula in terms of h only. We can do that using the given information that the height of the pile is always equal to the diameter.

If the diameter is equal to the height, then the radius is half the height; r = h/2. Then the volume formula is

Now we need to differentiate this formula to find the relationship between dh/dt and dV/dt.

(NOTE: Many beginning calculus students try to substitute the given value h=2 prior to this point. However, that gives a constant for the volume; and the derivative of a constant is 0 -- so we won't be able to finish the problem....)

We could plug in the given value h=2 at this point, along with the given value of 5 for dV/dt, to find the answer to the problem. But let's hold off on substituting the given value for h; perhaps later on we will be asked to find dh/dt when the height is some value other than 2.

So we are getting a formula for dh/dt in terms of dV/dt and h, so that we can, if required, find dh/dt for several different values of h.

Now we can plug in the given value of h to find dh/dt when the height is 2:

To 2 decimal places, dh/dt = 1.59 when h = 2.
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