Question

The radius of a circle is increasing at a constant rate of 6 meters per minute. At theinstant when the area of the circle is 81pi square meters, what is the rate of change ofthe area? Round your answer to three decimal places.

Answer 1

601.379 m²/min

Step-by-step explanation:

The area of the circle is calculated as

`A=\pi r^2`

Where r is the radius. To find the rate of change, we need to derivate the expression, so

`(dA)/(dt)=2\pi r\frac{\text{ dr}}{\text{ dt}}`

Now, we need to find r when the area is 81π², so solving the following equation for r, we get:

`\begin{gathered} 81\pi^2=\pi r^2 \\ (81\pi^2)/(\pi)=(\pi r^2)/(\pi) \\ 81\pi=r^2 \\ \sqrt[]{81\pi}=r \\ 9\sqrt[]{\pi}=r \end{gathered}`

Then, we can find dA/dt when the area is 81π², replacing r = 9√π and dr/dt by 6 m/min

`\begin{gathered} \frac{\text{ dA}}{\text{ dt}}=2\pi(9\sqrt[]{\pi})(6) \\ \frac{\text{ dA}}{\text{ dt}}=601.379^{}m^2\text{/min} \end{gathered}`

Therefore, the rate of change is 601.379 m²/min