Question

question 15 · 1 point the radius of a circle increases at a rate of 6 m/s. find the rate at which the area of the circle is increasing when the radius is 9 m. enter an exact answer in terms of π provide your answer below: the area of the circle is increasing at a rate of □ m²/s.

Explanation:

Step1: Recall the area formula of a 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: Identify given values

We are given that $r = 9$ m and $\frac{dr}{dt}=6$ m/s.

Step4: Substitute values into the derivative formula

Substitute $r = 9$ and $\frac{dr}{dt}=6$ into $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$. Then $\frac{dA}{dt}=2\pi\times9\times6$.

Step5: Calculate the result

$\frac{dA}{dt}=108\pi$ $m^{2}/s$.

Answer:

$108\pi$