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 πr3, where r is the radius of the hemisphere
Volume of the cone = 1/3πr2h, where r and h are the radius and height of the cone respectively.
Volume of the cylinder = πr2h, 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
= πr2H - 2/3πr3 - 1/3πr2h
= (1/3)πr2 (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 cm3
= 1131428.57/1000000 m3
= 1.13142657 m3
Thus, the volume of water left in the cylinder is 1.131 m3.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 13
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
Summary:
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. The volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm is 1.131 m3.
☛ Related Questions:
- A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.
- Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
- A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see Fig. 13.15).
- A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see Fig. 13.16).