Question

I want to know the steps of how to get the answer to the problem as well as the actual answer

Answer 1

$\bf \textit{area of a circle}\\\\ A=\pi r^2~~ \begin{cases} r=radius\\ -----\\ A=80\pi \end{cases}\implies 80\pi =\pi r^2 \implies \cfrac{80\pi }{\pi }=r^2 \\\\\\ 80=r^2\implies \sqrt{80}=r\\\\ -------------------------------$

$\bf \textit{area of a sector of a circle}\\\\ A=\cfrac{\theta r^2}{2}\qquad \begin{cases} r=radius\\ \theta =angle~in\\ \qquad radians\\ ------\\ A=36\pi\\ r=\sqrt{80} \end{cases}\implies 36\pi =\cfrac{\theta (\sqrt{80})^2}{2} \\\\\\ 72\pi =80\theta \implies \cfrac{72\pi }{80}=\theta \implies \cfrac{9\pi }{10}=\theta$

$\bf \textit{arc's length}\\\\ s=r\theta ~~ \begin{cases} r=radius\\ \theta =angle~in\\ \qquad radians\\ ------\\ r=\sqrt{80}\\ \qquad \sqrt{4^2\cdot 5}\\ \qquad 4\sqrt{5}\\ \theta =\frac{9\pi }{10} \end{cases}\implies s=4\sqrt{5}\cdot \cfrac{9\pi }{10}\implies s=\cfrac{18\pi \sqrt{5}}{5}$