Question

2. Find the volume of a cone formed from a sector with diameter of 72 inches and a central angle of 7pi/6 rad.

Answer 1

diameter= 72 in
radius=72/2=36 in
let h be height of cone.
then
\[\frac{ 36 }{ h }=\tan \frac{ 7\pi }{ 6 }\]
\[h=36 \cot \frac{ 7\pi }{ 6 }=36\cot (\frac{ \pi }{ 2 }+\frac{ 4\pi }{ 6})=-36\tan (\frac{ 2\pi }{ 3})=-36\tan (\pi- \frac{ \pi }{ 3 } )\]
\[h=36\tan (\frac{ \pi }{ 3 })=36\sqrt{3}\]
volume of cone\[=\frac{ 1 }{ 3}\pi r^2 h=\frac{ 1 }{ 3 }\pi (36)^2\times 36\sqrt{3}=15552\sqrt{3} \pi~ inch^3\]