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 of the area is Number
Units

Step 1 ：The problem involves a circle with a radius that is increasing at a rate of 10 centimeters per minute. We are asked to find the rate of change of the area when the radius is 5 centimeters.

Step 2 ：The area of a circle is given by the formula \(A = \pi r^2\), where \(r\) is the radius of the circle.

Step 3 ：We can solve this problem by taking the derivative of the area with respect to time. This gives us \(\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\).

Step 4 ：We are given that \(\frac{dr}{dt} = 10\) cm/min and \(r = 5\) cm.

Step 5 ：Substituting these values into the equation gives us \(\frac{dA}{dt} = 2\pi (5) (10)\).

Step 6 ：Solving this equation gives us \(\frac{dA}{dt} = 314.1592653589793\) cm²/min.

Step 7 ：Rounding to one decimal place, the rate of change of the area when the radius is 5 centimeters is approximately \(\boxed{314.2}\) square centimeters per minute.