Question

Determine the area of a cone that its volume is 36 cubic inches and the height is 9 inches.

Answer 1

`\bf \stackrel{\textit{volume of a cone}}{V=\cfrac{\pi r^2 h}{3}}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ V=36\\ h=9 \end{cases}\implies 36=\cfrac{\pi r^2(9)}{3}\implies 36\cdot \cfrac{3}{9}=\pi r^2 \\\\\\ 12=\pi r^2\implies \cfrac{12}{\pi }=r^2\implies \sqrt{\cfrac{12}{\pi }}=r \\\\[-0.35em] ~\dotfill`

`\bf \stackrel{\textit{surface area of a cone}}{SA=\pi r√(r^2+h^2)+\pi r^2}\implies SA=\pi \sqrt{\cfrac{12}{\pi }}\left( \sqrt{\cfrac{12}{\pi }+9^2} \right)+\pi \cdot \cfrac{12}{\pi } \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill SA\approx 68.54757~\hfill`