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

Radius of cone \(\displaystyle =5 \mathrm{~cm}\), height of cone \(\displaystyle =8 \mathrm{~cm}\) Volume of water in the cone \(\displaystyle =\frac{1}{3} \pi \times(5)^{2} \times 8\) \(\displaystyle =\frac{200 \pi}{3} \mathrm{~cm}^{3}\) Volume of water flows out \(\displaystyle =\frac{1}{4}\) (Volume of water in the cone) \(\displaystyle =\frac{1}{4} \times \frac{200 \pi}{3}=\frac{50 \pi}{3} \mathrm{~cm}^{3}\) Radius of sphere (lead shot) \(\displaystyle =0.5=\frac{1}{2} \mathrm{~cm}\) Volume of one lead shot \(\displaystyle =\frac{4}{3} \pi \times\left(\frac{1}{2}\right)^{3}\) \(\displaystyle =\frac{\pi}{6} \mathrm{~cm}^{3}\) Number of lead shots \(\displaystyle =\frac{\dfrac{50 \pi}{3}}{\dfrac{\pi}{6}}=100\)