When a right triangle of area 3 is rotated 360^ about its shorter… - Vidyadip

MCQ

When a right triangle of area $3$ is rotated $360^\circ $ 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
$0.99$
✓
$7.90$
C
$8.88$
D
$31.42$

✓

Answer

Correct option: B.

$7.90$

Let the length of shorter leg be hh, length of longer leg be $k$, volume of resulting solid is $30$.
The resulting solid is cone, so volume is $\frac{1}{3}\times\pi\text{k}^2\text{h}=30$
Area of triangle is $\frac{1}{2}\times\text{h}\text{k}=3,$ which implies $hk = 6$
which gives $\frac{1}{3}\times\pi6\text{k}=30,$ $\text{k}=\frac{15}{\pi}$
Volume when rotated with longer leg is
$\frac{1}{3}\times\pi\text{h}^2\text{k}=2\pi\text{h}$
$=\frac{2\pi6}{\text{k}}=\frac{12\pi}{15}\times\pi=7.9$

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