Question

a circle has an initial radius of 50 ft when the radius begins decreasing at the rate of 3 ft/min. what is the rate in the change of area at the instant that the radius is 27 ft? the rate of change of the area is (type an exact answer in terms of π.)

Explanation:

Step1: Write the formula for the area of a circle

The area formula of a circle is \(A=\pi r^{2}\), where \(A\) is the area and \(r\) is the radius.

Step2: Differentiate the area formula with respect to time \(t\)

Using the chain - rule \(\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}\).
Differentiate \(A = \pi r^{2}\) with respect to \(r\): \(\frac{dA}{dr}=2\pi r\).
We are given that \(\frac{dr}{dt}=- 3\) (negative because the radius is decreasing).
So \(\frac{dA}{dt}=2\pi r\cdot\frac{dr}{dt}\).

Step3: Substitute the given value of \(r\)

When \(r = 27\) and \(\frac{dr}{dt}=-3\), we substitute these values into the formula \(\frac{dA}{dt}=2\pi r\cdot\frac{dr}{dt}\).
\(\frac{dA}{dt}=2\pi\times27\times(-3)\).
\(\frac{dA}{dt}=- 162\pi\)

Answer:

\(-162\pi\) square feet per minute.