Related Rates

Added: 2026-05-22

[00:00:00]Hi, today we are looking at a related rates problem. We are given a screensaver on a computer screen shows a circle that expands outward. We know that at the instant when the circle has a radius of 9 cm, the radius is increasing at a rate of 3/2 cm per second. We need to find the rate at which the area of the circle is changing at that instant and we will have to indicate units of measure. We're talking about circles, so the formula for the area of a circle, we remember that because the radius is changing.

[00:00:31]Area is changing and that happens with respect to time. So, that's a classic related rates problem.

[00:00:38]We calculate the derivative of both sides with respect to time. So, dA/dt = π. Derivative of r² is 2r, but the radius also changes with respect to time. So, dA/dt = π * 2r dr/dt and then look at that. We know that the radius was 9 cm and we knew that the radius changes at a rate of 3/2 cm per second.

[00:00:59]So, now we just plug that in. dA/dt = π * 2 * 9 * dr/dt which is 3/2.

[00:01:09]Once we simplify that, we are going to get 9 * 3 27 π square centimeters per second. Right?

[00:01:18]We're multiplying centimeters * centimeters per second. We have centimeters squared per second. Indicate units of measure tells us that on AP test, we will be getting a point for the units. So, even if you are confused and you're not sure what to do, just think about the units, write down the units, at least you will get one point. Follow me for more calculus and I will see you next time.