question the area of a circle is increasing at a constant rate of 352 square feet per second. at the instant…

Explanation:

Step1: Recall area formula of circle

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 $t$

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

Step3: Solve for $\frac{dr}{dt}$

We know that $\frac{dA}{dt} = 352$ square feet per second and $r = 9$ feet. Rearranging the equation $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$ for $\frac{dr}{dt}$, we get $\frac{dr}{dt}=\frac{\frac{dA}{dt}}{2\pi r}$.

Step4: Substitute the given values

Substitute $\frac{dA}{dt}=352$ and $r = 9$ into the formula: $\frac{dr}{dt}=\frac{352}{2\pi\times9}=\frac{352}{18\pi}$.

Step5: Calculate the value

$\frac{352}{18\pi}=\frac{176}{9\pi}\approx6.217$.

Answer:

$6.217$