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

Conical vessel dimensions: height (h) = 8 cm, radius (r₁) = 5 cm. Lead shot radius (r₂) = 0.5 cm.

Let n be the number of lead shots dropped into the vessel.

Volume of water spilled equals the total volume of the dropped lead shots. The total volume of water overflown is \((\frac{50}{3})\pi\).

The volume of a single lead shot is \( (\frac{4}{3})\pi r^3 \), which calculates to \( (\frac{1}{6}) \pi \).

The number of lead shots (n) is calculated by dividing the total volume of water overflown by the volume of a single lead shot:

\(n = \frac{\text{Total volume of water overflown}}{\text{Volume of lead shot}} = \frac{ (\frac{50}{3})\pi }{(\frac{1}{6})\pi}\)

\(n = (\frac{50}{3})\times6 = 100\)