Answer:
601.379 m²/min
Explanation:
The area of the circle is calculated as
[tex]A=\pi r^2[/tex]Where r is the radius. To find the rate of change, we need to derivate the expression, so
[tex]\frac{dA}{dt}=2\pi r\frac{\text{ dr}}{\text{ dt}}[/tex]Now, we need to find r when the area is 81π², so solving the following equation for r, we get:
[tex]\begin{gathered} 81\pi^2=\pi r^2 \\ \frac{81\pi^2}{\pi}=\frac{\pi r^2}{\pi} \\ 81\pi=r^2 \\ \sqrt[]{81\pi}=r \\ 9\sqrt[]{\pi}=r \end{gathered}[/tex]Then, we can find dA/dt when the area is 81π², replacing r = 9√π and dr/dt by 6 m/min
[tex]\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}[/tex]Therefore, the rate of change is 601.379 m²/min
