Question

When a right triangle of area $3$ is rotated ${360}^o$ about its shorter leg, the solid that results has a volume of $30$. Calculate the volume of the solid if the same right triangle is rotated about its longer leg.

• A. 0.99
• B. 7.9
• C. 8.88
• D. 31.42
• E. 41.12

Answer 1

The correct option is B 7.9

Let the length of shorter leg be $h$, length of longer leg be $k$ , volume of resulting solid is $30$.
The resulting solid is cone, so volume is $\frac{1}{3}\times \pi k^{2}h=30$

Area of triangle is $\frac{1}{2}\times hk=3$ , which implies $hk=6$

which gives $\frac{1}{3}\times \pi 6k=30$ , $k=\frac{15}{\pi }$

Volume when rotated with longer leg is $\frac{1}{3}\times \pi h^{2}k=2\pi h=\frac{2\pi 6}{k}=\frac{12\pi }{15}\times \pi =7.9$