Question

Find the lateral area and surface area of a cone with an altitude of 5 feet and a slant height of 9 and 1/2 feet. Round to the nearest tenth, if necessary.

Answer 1

Check the picture below.

we know the LA is 9 and 1/2 or namely 19/2 and the height is 5, so

`\stackrel{slant~height}{\cfrac{19}{2}}~~ = ~~\stackrel{slant~height}{√(r^2+h^2)}\implies \left( \cfrac{19}{2} \right)^2~~ = ~~r^2+5^2\implies \cfrac{361}{4}~~ = ~~r^2+25 \\\\\\ \cfrac{361}{4} - 25~~ = ~~r^2\implies \cfrac{261}{4}=r^2\implies \sqrt{\cfrac{261}{4}}=r\implies \boxed{\cfrac{3√(29)}{2}=r} \\\\[-0.35em] ~\dotfill\\\\ LA=\pi r\stackrel{slant~height}{√(r^2+h^2)}\implies LA=\pi \left( \cfrac{3√(29)}{2} \right)\cfrac{19}{2}\implies \boxed{LA=\cfrac{57\pi √(29)}{4}}`

`~\dotfill\\\\ SA=\pi r√(r^2+h^2)~~ + ~~\pi r^2\implies SA=LA~~ + ~~\pi r^2 \\\\\\ SA=\cfrac{57\pi √(29)}{4}~~ + ~~\cfrac{3\pi √(29)}{2}\implies \boxed{SA=\cfrac{63\pi √(29)}{4}} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill LA\approx 241.1\qquad SA\approx 266.5~\hfill`