Question : An inverted conical–shaped vessel is filled with water to its brim. The height of the vessel is 8 cm and the radius of the open end is 5 cm. When a few solid spherical metallic balls each of radius $\frac{1}{2}$ cm are dropped in the vessel, 25% water is overflowed. The number of balls is:
Option 1: 100
Option 2: 400
Option 3: 200
Option 4: 150
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Correct Answer: 100
Solution :
Given: Height of the vessel = 8 cm
Radius = 5 cm.
Volume of the conical vessel = $\frac{1}{3}\pi r^2 h$
= $\frac{1}{3}\times \pi\times 5^2\times 8$
= $\frac{200}{3}\pi$ cm$^3$
The volume of 25% of water = $\frac{1}{4}\times\frac{200}{3}\pi=\frac{50}{3}\pi$ cm$^3$
The volume of the spherical metallic ball of the radius $R = \frac{4}{3} \pi R^3$
$=\frac{4}{3}\times \pi\times (\frac{1}{2})^3=\frac{\pi}{6}$ cm$^3$
The number of balls required = $\frac{\frac{50}{3}\pi}{\frac{\pi}{6}}$ = 100
Hence, the correct answer is 100.
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