Final answer:
The rate of change of the area of a circle with respect to the radius is 2πr. For a specific radius, this rate is calculated by substituting the value into the formula. For example, if r = 1.2 m, the rate of change is approximately 7.54 m²/m.
Step-by-step explanation:
The rate of change of the area of a circle with respect to the radius r can be found by differentiating the area formula A = πr² with respect to r. The derivative of the area A with respect to r is given by dA/dr = 2πr. This expression gives us the rate at which the area of a circle changes as the radius changes.
When looking for this rate for a specific value of r, you plug the value of r into the derivative equation. For example, if r = 1.2 m, then the rate of change of the area with respect to the radius is 2π(1.2 m) = 7.54 m²/m, approximately. This value represents how much the area of the circle will increase for each meter the radius is expanded.