### Video Transcript

A cylindrical tank with a radius of 90 centimeters is being filled with water so that the volume increases at a rate of 180 centimeters cubed per second. How fast is the height of the water increasing? The volume of the cylinder of height β is given by the formula π equals ππ squared β.

Letβs draw a picture using the information weβve been given. We have a cylindrical tank with a radius of 90 centimeters. We also know that the volume increases at a rate of 180 centimeters cubed per second. The question is how fast is the height of the water increasing. But what weβre really being asked is to find the rate of change of the height of the water. Letβs write down what we know already.

We know that the volume of a cylinder is ππ squared β. We know that the rate of change of volume dπ by dπ‘ is equal to 180 centimeters cubed per second, which is a constant. And we know that the radius of the cylinder is 90 centimeters. So π squared is 8100 centimeters. We can substitute this into the formula for volume which gives us π equals 8100πβ.

Now, remember, we want to find the rate of change of the height β. And we know that both volume and height are functions of time. They both change over time as water is poured into the cylinder. So if we differentiate volume with respect to time, as volume is a function of height, we also need to differentiate height with respect to time. So dπ by dπ‘ is 8100π times dβ by dπ‘.

We know that the volume changes at a constant rate of 180 centimeters cubed per second. So dπ by dπ‘ is 180. And thatβs equal to 8100π times dβ by dπ‘. So now to find dβ by dπ‘ which is actually what weβre looking for, we make dβ by dπ‘ the subject of this equation. Dividing both sides by 8100π leaves dβ by dπ‘ on the right-hand side. Cancelling this down, we find dβ by dπ‘ is one over 45π. Remember, volume is measured in cubic centimeters, but height is measured in centimeters. So dβ by dπ‘ is one over 45π centimeters per second.