# A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm

**Solution:**

A figure is drawn below in accordance with the given information to visualize effectively.

From the figure, we can observe that the solid fits totally into the cylinder as the height of the solid is the same as the height of the cylinder.

Since the cylinder was full of water before immersing the solid into it, so the entire space in the cylinder is occupied by the solid and the left-over water.

Volume of the cylinder = volume of the solid + volume of the water left in the cylinder

Therefore,

Volume of the water left in the cylinder = volume of the cylinder - volume of the solid

Hence,

Volume of the water left in the cylinder = volume of the cylinder - volume of the hemisphere - volume of the cone

Let us find the volume of the solid by using formulae;

Volume of the hemisphere = 2/3 πr^{3}, where r is the radius of the hemisphere

Volume of the cone = 1/3πr^{2}h, where r and h are the radius and height of the cone respectively.

Volume of the cylinder = πr^{2}h, where r and h are the radius and height of the cylinder respectively.

Radius of hemispherical part = Radius of conical part = Radius of cylinder = r = 60 cm

Height of the conical part of solid = h = 120 cm

Height of the cylinder, = H = 180 cm

Volume of the water left in the cylinder = volume of the cylinder - volume of the hemisphere - volume of the cone

= πr^{2}H - 1/3πr^{2}h - 2/3πr^{3}

= 1/3πr^{2 }(3H - h - 2r)

= 1/3 × 22/7 × 60 cm × 60 cm × (3 × 180 cm - 120 cm - 2 × 60 cm)

= 1/3 × 22/7 × 60 cm × 60 cm × 300 cm

= 7920000/7 cm^{3}

= 1131426.57/1000000 m^{3}

= 1.13142657 m^{3}

Thus, the volume of water left in the cylinder is 1.131 m^{3}.

**Video Solution:**

## A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm

### NCERT Solutions for Class 10 Maths - Chapter 13 Exercise 13.2 Question 7 :

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm

The volume of water left in the right circular cylinder of radius 60 cm and height 180 cm is 1.13142657 m^{3}