Question

A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of the balls are 1.5 cm and 2 cm. Find the diameter of the third ball.

Answer 1

The radius of spherical ball = 3 cm.

Volume of spherical ball = `4/3` πr³

= `4/3` π × 3 × 3 × 3

= 36 π cm³

∵ Volume of a spherical ball = Total volume of three small spherical balls
∵ The radii of the balls are 1.5 cm and 2 cm.
∴ Let the radius of the third ball = r
∴ Volume of a spherical ball = Total volume of three small spherical balls.

36π = `4/3π (3/2)^3 + 4/3 π (2)^3 + 4/3 π r^3`

36π = `4/3π xx 27/8 + 4/3 π xx 8 + 4/3 π r^3`

36π = `4/3π ( 27/8 + 8 + r^3 )`

`(36π xx 3)/(4π)`

= `27/8 + 8 + r^3`

27 = `(27 + 64)/8 + r^3`

27 = `91/8 + r^3`

`27 - 91/8 = r^3`

`(216 - 91)/8 = r^3`

`125/8 = r^3`

r = `root(3)(125/8)`

r = `5/2` cm.
The diameter of the third ball = 2r = 2 x `5/2` = 5 cm.