Calculus Problem: Please Help!!

Question

The height of a cylinder with a fixed radius of 10 cm is increasing at the rate of 0.5 cm/min. Find the rate of change of the volume of the cylinder (with respect to time) when the height is 30cm.

Options:

1. 50π
2. 100π
3. 150π
4. None of these

Answer 1

The volume of a cylinder can be written as V = πr²h. In this case, r is a fixed number of 10 so we can re-write this volume equation as V = 100πh and since the only variable is "h", this becomes:

V(h) = 100πh

We can now take the derivative of this function with respect to time:

We are told that dh/dt (the rate that the height is changing) is 0.5cm/min so we can plug that in giving us:

dV/dt = 100π(0.5) = 50π cm³/min (notice that it doesn't matter that the height is 30)