Final answer:
The rate of change of the area of a circle with respect to the radius can be found using calculus. The formula for the rate of change is 2π times the radius. When the radius is 3, the rate of change is 6π.
Step-by-step explanation:
The rate of change of the area of a circle with respect to the radius, also known as the derivative of the area with respect to the radius, can be found using calculus. The formula for the area of a circle is A = πr², where r is the radius. To find the rate of change of the area with respect to the radius, we differentiate this formula with respect to r.
Using the power rule of differentiation, we get dA/dr = 2πr. This means that the rate of change of the area of a circle with respect to the radius is 2π times the radius.
Substituting r = 3 into the formula, we get dA/dr = 2π(3) = 6π. Therefore, the rate of change of the area of a circle with respect to the radius when r = 3 is 6π.