Question

The radius of a circle is increasing at a rate of 10 centimeters per minute. Find the rate of change of the area when the radius is 5 centimeters.

Round your answer to one decimal place.
The rate of change is the area is: [Number][Units]

Answer 1

The rate of change of the area when the radius is 5 centimeters is 100π cm²/min (approximately 314.2 cm²/min when rounded to one decimal place).

Step-by-step explanation:

To find the rate of change of the area when the radius is 5 centimeters, we can use the formula for the area of a circle, which is A = πr². Since the radius is increasing at a rate of 10 centimeters per minute, we need to find the derivative of the area with respect to time.

Let's differentiate the formula for the area of the circle with respect to time:

dA/dt = 2πr(dr/dt)

Substituting the given values, dA/dt = 2π(5)(10) = 100π cm²/min.

Therefore, the rate of change of the area when the radius is 5 centimeters is 100π cm²/min, which is approximately 314.2 cm²/min when rounded to one decimal place.