# 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 = pir^2h and I can determine the rate of change in Volume is dV/dt=pir^2dh/dt and the rate of change of height is dh/dt=1/pir^2 yimes dV/dt Using that formula I can determine that the water is rising at a rate of 3/25pi m/min. But I cannot seem to figure out how to factor in height so that it is half full. Or is this wrong?

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 = $\pi {r}^{2}h$ and I can determine the rate of change in Volume is $dV/dt=\pi {r}^{2}dh/dt$ and the rate of change of height is $dh/dt=1/\pi {r}^{2}×dV/dt$
Using that formula I can determine that the water is rising at a rate of $3/25\pi$ m/min. But I cannot seem to figure out how to factor in height so that it is half full. Or is this wrong?
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niveaus7s
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.