Question

A vessel is in the form of an inverted cone. Its height is $8$cm and the radius of its top, which is open, is $5$cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius $0.5$cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

Answer 1

Volume of water in a Cone,
$=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\pi \times (5{)}^2\times 8$
$=\frac{200}{3}\pi c{m}^2$
Volume of lead shot $=\frac{4}{3}\pi r^3$
$=\frac{4}{3}\pi \times (0.5{)}^3$
$=\frac{\pi }{6}c{m}^3$
Let the number of lead balls kept in vessel $\frac{1}{4}$ of the water flows out,
$\Rightarrow n\times \frac{\pi }{6}=\frac{1}{4}\times \frac{200\pi }{3}$
$\Rightarrow n\times \frac{\pi }{6}=\frac{100}{6}\pi$
$\therefore n=100$
$\therefore$ Number of lead balls = 100