Cylindrical tank rate of change

Question

Water is pouring into a cylinder with a radius of 5m and height of 20m at a rate of 3 cubic metres a minute. Find the rate of change of height when the tank is half full.

Now the Volume V = $πr^2h$ and I can determine the rate of change in Volume is $dV/dt = πr^2dh/dt$ and the rate of change of height is $dh/dt = 1/πr^2\times dV/dt$

Using that formula I can determine that the water is rising at a rate of $3/25π$ m/min. But I cannot seem to figure out how to factor in height so that it is half full. Or is this wrong?

Answer 1

You have correctly identified that water level is rising at the rate of $\frac{dh}{dt}=\frac{3}{25\pi}$ m/min.

Observe that $\frac{3}{25\pi}$ does not depend on the implicit variable $t$ (the time in minutes), therefore the rate of change of height is the same when the tank is half full or full. You've already found it.

Answer 2

Hint: You have determined that the rate of change of height with time, at any time-point, is constant. What does that tell you about the specific point in time when the tank is half-full -- or quarter-full, etc., for that matter?