A circle's radius is increasing at 3 cm/s. How fast is its area increasing when the radius is 5 cm?

Step 1: Identify the known rate and the rate we need.

The radius is increasing at:
dr/dt = 3 cm/s

We want to find the rate of change of the area:
dA/dt

when:
r = 5 cm

Step 2: Write the formula for the area of a circle.

A = πr^2

Step 3: Differentiate both sides with respect to time t.

dA/dt = 2πr(dr/dt)

This equation relates the rate of change of the area to the rate of change of the radius.

Step 4: Substitute the known values.

r = 5
dr/dt = 3

So:
dA/dt = 2π(5)(3)

Step 5: Simplify.

dA/dt = 30π

Therefore, when the radius is 5 cm, the area is increasing at 30π square centimeters per second.