1. Conversion of Solid from One Shape to Another
We all have over head tanks in our house which get filled by inlet pipes. Suppose if we want to know how much time it will take to fill the tank, how do we do those calculations rather than actually measuring the time it takes to fill?
If we know the dimensions of the tank and the pipe and the quantity of water that flows through the pipe in a given time, we can easily determine the time it takes to fill the tank. This is based on the principle that volume remains the same when a liquid is transferred from one container to another or a solid is converted from one form to another.
Conversion of solids from one shape to another
The total volume remains the same:
(i) When solids are converted from one form to another
(ii) liquids are poured from one container to another
Example
- 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
- Volume of the cone = Volume of the sphere
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- r = 6 cm
- \
2. Surface Area of a Combination of Solids
Suppose you want to paint a granary which has in the shape of a cylinder with a hemispherical dome, how much paint would you need? These are practical problems which can be addressed by computing the surface area of individual shapes.┬а
Surface area of a combination of hemisphere and cone
TSA of new solid = CSA of the hemisphere + CSA of the cone
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- r = radius
- l = slant height of the cone
Surface area of a combination of two hemispheres and a cylinder
The total surface area of a combination of solid is equal to the total curved surface area visible to our eyes.
In this case, the total surface area of the new solid is the sum of the curved surface areas of each of the individual parts
TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere
- TSA = total surface area
- CSA = curved surface area
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3. Volume of a Combination of Solids
You have a composite shape. Perimeter of composite shapes is not equal to the perimeters of the individual shapes but area of composite shapes is equal to the area of the individual shapes.
What happens in the case of volume? Will it be equal to volume of individual shapes?
Volume of combination of solids
The volume of the solid formed by joining two basic solids will actually be the sum of the volumes of its constituents.
For example, Total volume of the solid = Volume of the cuboid + Volume of the half cylinder
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When a cone is cut into two, we donтАЩt get 2 cones. We get a cone and a frustum. A bucket looks like a frustum.
Frustrum of a cone
Slice (or cut) through the cone with a plane parallel to its base and then remove the cone that is formed on upper side of that plane. The lower portion that is left over is called the frustum of the cone.
- Figure (a): we slice (or cut) through the cone with a plane parallel to its base
- Figure (b): remove the cone that is formed on one side of the plane
- Figure (c): the part that is now left over on the other side of the plane is called the frustum of the cone
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- Volume of the frustum of a cone =┬а
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