Question

Three cylinders each of height $16$ cm and radius of base $4$ cm are placed on a plane, so that each cylinder touches the other two. Then the volume of region enclosed between the three cylinders in $c{m}^3$ is

• A. $98(4\sqrt{3}-\pi )$
• B. $98(2\sqrt{3}-\pi )$
• C. $98(\sqrt{3}-\pi )$
• D. $128(2\sqrt{3}-\pi )$

Answer 1

Answer: (D)

$128(2\sqrt{3}-\pi )$

The required area $=$ Area of $\Delta ABC$ $-$ $3\times$ Area of ${60}^{\circ }$ sectors

$=\left(\frac{\sqrt{3}}{4}\times 8^2\right)-(3\times \frac{60}{360}\times \pi 4^2)$

$=16\sqrt{3}-8\pi =8(2\sqrt{3}-\pi )c{m}^2$

Required volume $=$ Area of region $\times$ height

= $16\times 8(2\sqrt{3}-\pi )$

$=128(2\sqrt{3}-\pi )c{m}^3$