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Surface Areas and Volumes

We know how to find the curved surface areas of three-dimensional figures such as cone, cylinder, sphere, etc. Now, we will learn to find out the curved surface area of a combination of solids.

Let us consider the following figure.

The given figure shows a temple. In this figure, the base of the temple is a right circular cylinder and the top is a right circular cone. The height of the conical part is half the height of the cylindrical part and the radius of the base of the temple is 2 ft more than the height of the conical part. The temple has to be plastered from outside at the rate of Rs 5 per square foot.

Can we find the approximate cost of plastering the temple from outside?

In order to find the cost, we have to find the curved surface area of the temple. By adding the curved surface areas of cone and cylinder, we can obtain the curved surface area of the temple.

First of all, let us review the formulae to find the curved surface areas of different solids. The formulae for different solids are given in the following table.

 Name of Solid Figure Formula Cuboid Lateral or curved surface area = 2h (l + b) where l = length, b = breadth, and h = height of the cuboid Cube Lateral or curved surface area = 4l2 where l = length of the edge of the cube Right Circular Cylinder C.S.A. = 2πrh where r = radius and h = height of the cylinder Right Circular Cone C.S.A. = πrl where r = radius, l = = slant height, and h = height of the cone Sphere and Hemisphere C.S.A. of sphere = 4πr2 C.S.A. of hemisphere = 2πr2 where r = radius of the sphere

Now let us look at the video to understand the solution to the problem discussed in the starting.

In this way, we can find the curved surface area of a combination of solids. Now, let us solve some more examples to understand the concept better.

Example 1:

A toy is in the form of a hemisphere mounted by a cone. The diameter of the hemisphere is 21 cm and height of the whole toy is 24.5 cm. If the surface of the toy is painted at the rate of Rs 1 per 5 cm2, then find the cost required to paint the entire toy.

Solution:

It is given that the diameter of the hemisphere is 21 cm.

∴ Radius of the hemisphere, r = 10.5 cm