Question

A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. One-fourth of the water flows out of the vessel. Find the radius of a spherical ball ?

Answer 1

Height of conical vessel, h = 8 cm
Radius of conical vessel, r = 5 cm
Volume of conical vessel, V \[= \frac{1}{3}\pi r^2 h\]

\[= \frac{1}{3}\pi \left( 5 \right)^2 \times 8\]
\[ = \frac{200\pi}{3} {cm}^3\]

When 100 spherical lead balls are dropped into the vessel, one-fourth of the water flows out of the vessel.
Let R be the radius of a spherical ball.
∴∴ 100 × Volume of one spherical lead ball = `1/4 `x Volume of vessel

\[\Rightarrow 100 \times \frac{4}{3}\pi R^3 = \frac{1}{4} \times \frac{200\pi}{3}\]

\[ \Rightarrow R^3 = \frac{1}{4} \times \frac{200\pi}{3} \times \frac{3}{100 \times 4\pi}\]

\[ \Rightarrow R^3 = \frac{1}{8}\]

\[ \Rightarrow R = \frac{1}{2} = 0 . 5 cm\]

Hence, the radius of each spherical ball is 0.5 cm