# Cylindrical tank rate of change

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?

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.