question the radius of a circle is decreasing at a constant rate of 3 inches per minute. at the instant when…

Explanation:

Step1: Recall the area formula

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

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 $\frac{dr}{dt}=- 3$ inches per minute (negative because the radius is decreasing) and $r = 8$ inches.

Step4: Substitute values

Substitute $r = 8$ and $\frac{dr}{dt}=-3$ into the equation $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$. So, $\frac{dA}{dt}=2\pi\times8\times(-3)$.

Step5: Calculate the result

$\frac{dA}{dt}=-48\pi\approx - 150.796$ square inches per minute.

Answer:

$-150.796$