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Question

(A) The volume of the conical vessel is given by $V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} 	imes \pi 	imes 5^2 	imes 8 = \frac{200\pi}{3}\, c

Vedclass · Accepted Answer

$100$

Vedclass · Answer

(A) The volume of the conical vessel is given by $V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} 	imes \pi 	imes 5^2 	imes 8 = \frac{200\pi}{3}\, cm^3$.The volume of water that overflows is $25\%$ of the total volume of the cone,which is equal to the total volume of the spherical balls dropped into the vessel.Volume of water overflowed $= \frac{1}{4} 	imes V_{cone} = \frac{1}{4} 	imes \frac{200\pi}{3} = \frac{50\pi}{3}\, cm^3$.The volume of one spherical ball with radius $r_b = \frac{1}{2}\, cm$ is $V_{ball} = \frac{4}{3} \pi r_b^3 = \frac{4}{3} 	imes \pi 	imes (\frac{1}{2})^3 = \frac{4}{3} 	imes \pi 	imes \frac{1}{8} = \frac{\pi}{6}\, cm^3$.Let $n$ be the number of balls. Then,$n 	imes V_{ball} = 	ext{Volume of water overflowed}$.$n 	imes \frac{\pi}{6} = \frac{50\pi}{3}$.$n = \frac{50\pi}{3} 	imes \frac{6}{\pi} = 50 	imes 2 = 100$.Therefore,the number of balls is $100$.

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