Question

Question 5
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

Radius of cone = 5 cm, height of cone = 8 cm, radius of sphere = 0.5 cm
Volume of cone
$=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\pi \times 5^2\times 8$
$=\frac{200}{3}\pi c{m}^3$
Volume of lead shot
$=\frac{4}{3}\pi r^3$
$=\frac{4}{3}\pi \times {0.5}^3$
$=\frac{1}{6}\pi c{m}^3$

Given that, $\text{number of lead shots}\times \left(\text{volume of the lead shot}\right)=\frac{1}{4}\times \left(\text{volume of the cone}\right)$
$\therefore \text{Number of lead shots}=(\frac{1}{4}\times (\text{volume of the cone}\left)\right)÷\left(\text{volume of the lead shot}\right)$
$=(\frac{1}{4}\times \frac{200}{3}\pi )÷\frac{1}{6}\pi$
$=\frac{50\pi }{3}\times \frac{6}{\pi }=100$