Question

a circle has an initial radius of 50 ft when the radius begins decreasing at the rate of 4 ft/min. what is the rate in the change of area at the instant that the radius is 24 ft? write an equation relating the area of a circle, a, and the radius of the circle, r. (type an exact answer, using $pi$ as needed.)

Explanation:

Step1: Recall area - radius formula

The formula for the area of a circle is $A=\pi r^{2}$.

Step2: Differentiate with respect to time

Differentiate both sides of the equation $A = \pi r^{2}$ with respect to time $t$ using the chain - rule. $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$.

Step3: Identify given values

We are given that $\frac{dr}{dt}=- 4$ ft/min (negative because the radius is decreasing) and $r = 24$ ft.

Step4: Substitute values into the derivative equation

Substitute $r = 24$ ft and $\frac{dr}{dt}=-4$ ft/min into $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$. Then $\frac{dA}{dt}=2\pi\times24\times(-4)$.

Step5: Calculate the value of $\frac{dA}{dt}$

$\frac{dA}{dt}=-192\pi$ square - feet per minute.

For the first part of the question (writing the equation relating the area and the radius):

Answer:

$A=\pi r^{2}$

For the second part (finding the rate of change of the area):