Question

the radius ( r ) of a circle is increasing at a rate of 3 centimeters per minute. find the rate of change of the area (in ( mathrm{cm}^{2} / mathrm{min} )) when ( r = 30 ) centimeters. ( mathrm{cm}^{2} / mathrm{min} )

Explanation:

Step1: Write the formula for the area of a circle

The area of a circle is \(A=\pi r^{2}\).

Step2: Differentiate both sides with respect to time \(t\)

Using the chain - rule, \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\).

Step3: Substitute the given values

We are given that \(\frac{dr}{dt} = 3\) cm/min and \(r = 30\) cm.
Substitute these values into the equation \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\):
\(\frac{dA}{dt}=2\pi\times30\times3\).

Step4: Calculate the value

\(\frac{dA}{dt}=180\pi\approx 180\times 3.14 = 565.2\)

Answer:

\(565.2\)