Question

The area of a circle is increasing at a constant rate of 88 square meters per second. At the instant when the area of the circle is 100π square meters, what is the rate of change of the radius? Round your answer to three decimal places.

Answer 1

To find the rate of change of the radius when the area of a circle is increasing at a constant rate, we can differentiate the formula for the area of a circle with respect to time and solve for the rate of change of the radius.

Step-by-step explanation:

To find the rate of change of the radius, we can differentiate the formula for the area of a circle with respect to time. The area of a circle is given by A = πr², where r is the radius. Differentiating this equation with respect to time, we get dA/dt = 2πr(dr/dt).

Given that dA/dt = 88 square meters per second and at the instant when the area is 100π square meters, we can substitute these values into the equation and solve for dr/dt.

Substituting, we have 88 = 2π(r)(dr/dt), and when A = 100π, r = 10.

Plugging in the values, we get 88 = 2π(10)(dr/dt), which simplifies to dr/dt = 88/(20π).

Rounding to three decimal places, the rate of change of the radius is approximately 0.698 meters per second.