How do I set up the equation and get the answer?

Question

The radius of a circle is increasing at a rate of

2 meters per minute.

When the radius is 13 meters,

then how fast is the AREA changing?

The rate of change of the AREA is  square meters per minute.

Answer 1

This is a related rates problem. Both the radius and the area are changing with respect to time. Assuming that area is a function of time and radius is a function of time, finding the derivative requires application of the chain rule.

The formula for area of a circle is:

A = πr².

When we take the derivative with respect to time (t) we gat this:

dA/dt = 2πr dr/dt (the dr/dt is a result of using the chain rule, "multiply by the derivative of the inside" because r is a function of t).

We know dr/dt is 2 m/min, and we want to find dA/dt at the point in time when r = 13 m. Just substitute:

dA/dt = 2π(13 m)(2 m/min )

= 52π m²/min