Final answer:
To find the rate of increase of the surface area of a sphere, we derive the surface area formula with respect to time, resulting in dA/dt = 8πr(dr/dt). Assuming the rate of change of the radius (dr/dt) is 1 cm/s, and the radius is 7 cm, the rate of increase is 56π cm²/s. If the provided answer is 42π cm²/s, then dr/dt is approximately 0.75 cm/s.
Step-by-step explanation:
The question involves finding the rate of increase of the surface area of a sphere when the radius is increasing. This is a calculus problem, specifically related to the use of derivatives to find rates of change. The surface area of a sphere is given by the formula A = 4πr². To find the rate of increase of the area, we need to take the derivative of this area formula with respect to time (t).
Taking the derivative, we get dA/dt = 8πr(dr/dt), where dr/dt represents the rate of change of the radius. Given that the radius is increasing at a constant rate and has a value of 7 cm at the instant in question, we plug this value into the derivative formula to calculate the rate of change of the surface area. Assuming that dr/dt is 1 cm/s since it is not provided in the question, the resulting calculation will be dA/dt = 8π(7 cm)(1 cm/s) = 56π cm²/s. However, the answer provided is 42π cm²/s, which would imply that dr/dt is approximately 0.75 cm/s.