Oil leaks out from a damaged oil tanker and forms a circle around the tanker. If the radius of the circle of oil is increasing at 25m/min, find the rate of change of the area when the radius is 20km

A = πr^2

dA/dt = 2πr dr/dt
when r = 20000 m , dr/dt = 25 m/min

dA/dt = 2π(20000)(25) = 1,000,000π m^2/min
= π km^2/min

(1 km = 1000 m
1 km^2 = (1000)^2 or 1,000,000 m^2)

To find the rate of change of the area of the oil spill, we can differentiate the equation for the area of a circle with respect to time.

The formula for the area of a circle is: A = πr^2, where A represents the area and r represents the radius.

Differentiating both sides of the equation with respect to time (t) gives us:

dA/dt = d/dt (πr^2)

The derivative of r^2 with respect to t is 2r(dr/dt) using the chain rule. Therefore, the equation becomes:

dA/dt = 2πr(dr/dt)

Now, we are given that the radius of the circle is increasing at a rate of 25 m/min. So, dr/dt = 25 m/min.

We are required to find the rate of change of the area when the radius is 20 km. However, the units of measurement for the rate of change of radius and the given radius of 20 km do not match. Therefore, we need to convert the units to ensure consistency.

1 km = 1000 m. Therefore, 20 km = 20 × 1000 = 20000 m.

Substituting the values into the equation:

dA/dt = 2πr(dr/dt)
dA/dt = 2π(20000)(25)
dA/dt = 1000000π

Hence, the rate of change of the area of the oil spill when the radius is 20 km is 1000000π square meters per minute.