the radius ( r ) of a circle is increasing at a rate of 3 centimeters per minute. find the rate of change of…

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} )

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} )

Answer

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)