Question

A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into vessel. One-fourth of the water flows out of the vessel. Find the radius of a spherical ball.

Answer 1

Height of conical vessel, h = 8 cm

Radius of conical vessel, r = 5 cm

Volume of conical vessel, V = \(\frac 13 \pi r^2 h\)

\(= \frac 13 \pi (5)^2 \times 8\)

\(= \frac{200 \pi }3 \) cm³

When 100 spherical lead balls are dropped into the vessel, one-fourth of the water flows out of the vessel.

Let R be the radius of a spherical ball.

∴ 100 × Volume of one spherical lead ball = \(\frac 14\) x Volume of vessel

⇒ \(100 \times \frac 43 \pi R^3 = \frac 14 \times \frac{200 \pi }3\)

⇒ \(R^3 = \frac 14 \times \frac{200 \pi}3 \times \frac 3 {100 \times 4\pi}\)

⇒ \(R^3 = \frac 18\)

⇒ \(R = \frac 12 = 0.5 \) cm

Hence, the radius of each spherical ball is 0.5 cm.

Answer 2

Volume of water in cone