Question

A ${30}^0-{60}^0-{90}^0$ triangle has the smallest side equal to 10 cm. This triangle is first rotated about the smallest side and then about the second largest side. If the volumes of the cones generated is $a$ and $b$ respectively, then -

• A. $a<b$
• B. $a=b$
• C. $a>b$
• D. $a=2b$

Answer 1

The correct option is C $a>b$

The smallest sides are 10cm. and $10\sqrt{3}cm.$
When triangle is rotated about the 10 cm side:
radius $=10\left(\sqrt{3}\right)$ and height$=10cm$
volume, $a=\frac{1}{3}\pi \left(10\sqrt{3}{)}^2\right(10)c{m}^3=1000\pi$
When triangle is rotated about $10\sqrt{3}cm.$:
Here, radius$=10cm$ and height$=10\sqrt{3}cm.$
volume, $b=\frac{1}{3}\pi \left({10}^2\right)\left(10\sqrt{3}\right)c{m}^3=\frac{1000\pi \sqrt{3}}{3}$
Clearly, $a>b$