In this blog, we will discuss the conversion of solid from one shape to another. Fatima was helping her mother in serving food. She inverted the bottle filled with water in the jug. Does the quantity of water change? Of course not. Ok, let us take some clay. Let us mold it into a ball. Now I want to reshape it and make a star of the same clay without adding or taking out clay from it.
The volume of the new shape that is star will be the same as the volume of the earlier shape that is the ball. This is what we have to remember when we come across objects which are converted from one shape to another, or when a liquid that originally filled one container of a particular shape is poured into another container of a different shape or size.
Every solid occupies some volume. When one solids shape is converted to another, its volume remains the same despite the difference in shape formed. We need to remember this while calculating the volume after the conversion of solid from one shape to another.
To understand how to calculate the volume after the conversion of solid from one shape to another, let us consider some examples.
1. A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere.
Solution : Volume of cone = 1 /3 x π x 6 x 6 x 24cm3
Let r is the radius of the sphere, then its volume is 4/3πr3.
The volume of clay in the form of the cone = the volume clay in the form of the sphere
4/3πr3 = 1/3 x π x 6 x 6 x 24cm3
r3= 3 x 3 x 24 = 33 x 23
Therefore, the radius of the sphere is 6 cm.
2. Selvi’s house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (an underground tank) which is in the shape of a cuboid. The sump has dimensions 1.57 m × 1.44 m × 95cm. The overhead tank has its radius of 60 cm and a height of 95 cm. Find the height of the water left in the sump after the overhead tank has been completely filled with water from the sump which had been full. Compare the capacity of the tank with that of the sump.
Solution: The volume of water in the overhead tank = the volume of the water removed from the sump.
The volume of water in the overhead tank (cylinder) = πr2h = 3.14 x 0.6 x 0.6 x 0.95 m3.
The volume of water in the sump when full = l x b x h = 1.57 x 1.44 x 0.95 m3.
The volume of water left in the sump after filling the tank = [(1.57 x 1.44 x 0.95) – (3.14 x 0.6 x 0.6 x 0.95)] m3
= (1.57 x 0.6 x 0.6 x 0.95 x 2) m3
So, the height of the water left in the sump = volume of water left in the sump / l x b.
= (1.57 x 0.6 x 0.6 x 0.95 x 2) /1.57 x 1.44.
= 0.475 m = 47.5 cm.
Also, Capacity of tank / Capacity of sump
=3.14 × 0.6 X 0.6 x 0.95 / 1.57 × 1.44 × 0.95
Therefore, the capacity of the tank is half the capacity of the sump.