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.

The figure drawn below of the vessel and lead shot is to visualize it.

As the water is filled up to the brim in the vessel.

The volume of water in the vessel = Volume of the conical vessel

On dropping a certain number of lead shots sphere into the vessel one-fourth of the water flows out.

The volume of all lead shots dropped into the vessel = 1/4 × Volume of the water in the vessel

Hence,

Number of lead shots = 1/4 × volume of the water in the vessel ÷ volume of each lead shot

We will find the volume of the water in the vessel and lead shot by using formulae;

Volume of  sphere = 4/3 πr³

where r is the radius of the sphere

Volume of  cone= 1/3 πR²h

where R and h are the radius and height of the cone respectively

Height of the conical vessel, h = 8 cm

Radius of the conical vessel, R = 5 cm

Radius of the spherical lead shot, r = 0.5 cm

Number of lead shots = 1/4 × volume of the water in the vessel ÷ volume of each lead shot

= 1 /4 × (1/3) πR²h × 3/4πr³

= πR²h/12 × 3/4πr³

= R²h / 16r³

= (5cm × 5 cm × 8 cm) / (16 × 0.5 cm × 0.5 cm × 0.5 cm)

= 100

Thus, the number of lead shots dropped in the vessel is 100.