A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of the top which is open is 5 cm. It is filled with water up to the rim. 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.

Given,

Height of cone (H) = 8 cm

Radius of cone (R) = 5 cm

Radius of sphere (r) = 0.5 cm.

Let no. of lead shots dropped in vessel be n.

According to question,

One-fourth of the water flows out after dropping lead shots.

∴ $\dfrac{1}{4}$ x Volume of cone = n × Volume of each sphere

$\Rightarrow \dfrac{1}{4} \times \dfrac{1}{3}πR^2H = n \times \dfrac{4}{3}πr^3 \\[1em] \Rightarrow \dfrac{R^2H}{4} = n \times 4r^3 \\[1em] \Rightarrow n = \dfrac{R^2H}{16r^3} \\[1em] \Rightarrow n = \dfrac{5^2 \times 8}{16 \times (0.5)^3} \\[1em] \Rightarrow n = \dfrac{25 \times 8}{16 \times 0.125} \\[1em] \Rightarrow n = \dfrac{200}{2} \\[1em] \Rightarrow n = 100.$

Hence, no. of shots dropped = 100.