Question

A cylinder of radius R is filled with water to its brim. A sphere of the same radius is drowned fully into water. If the volume of water left in the cylinder is equal to half of the volume of the sphere, what is the height of the cylinder?​

Answer 1

Given,

The radius of the cylinder = R

The radius of the sphere = R

Volume of the remaining water = Half of the volume of the sphere

To find,

The height of the cylinder.

Solution,

Volume of the sphere = 4/3 πR³ unit³

Displaced water by the sphere = 4/3 πR³ unit³

(According to the Archimedes principle.)

Remaining water = 4/3 πR³ ÷ 2 = 4/3 πR³ × 1/2 = 2/3 πR³ unit³

Total volume of the cylinder

= Displaced water + Remaining water

= 4/3 πR³ + 2/3 πR³

= πR³ (4/3+2/3)

= πR³ (4+2/3)

= πR³ (6/3)

= 2πR³ unit³

Let, the height of the cylinder = h unit

Volume of the cylinder = πR²h unit³

According to the data mentioned in the question,

πR²h = 2πR³

h = 2πR³/πR²

h = 2R

Hence, the height of the cylinder is 2R units.