Construct a cylinder and a cone of the height 30 cm and radius 15 cm. Prove that the volume of the cylinder is three times that of the cone.

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A cylinder that has a radius r and a height h is made up of a large number of circular discs, each with a radius of r. Area of each such disc is `pir^2`  and to calculate the volume of the entire cylinder, we can integrate this area over the...

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A cylinder that has a radius r and a height h is made up of a large number of circular discs, each with a radius of r. Area of each such disc is `pir^2`  and to calculate the volume of the entire cylinder, we can integrate this area over the total cylinder height as:

Volume = `int(pir^2)dh= pir^2h`

In case of a cone, it is made up of a number of discs with uniformly decreasing radius. The radius of the disc at the base is r and it reduces as we move towards its top. If the slope of the side of the cone is denoted as s (= r/h), then each disc has a radius of sh' (where h' is the height of that particular disc from one end of the cone).

The volume of the cone is the integral of areas of all such discs over the height h. That is,

Volume = `intpi(sh')^2dh' = pis^2(h^3)/3 = pi(r/h)^2 (h^3)/3 = 1/3 pir^2h`

Thus, we can see that the volume of a cone is one third of that of a cylinder. 

For the given case of radius 15 cm and height 30 cm, volume of cylinder is 21,205.8 cm^3 and that of the cone is 7,068.6 cm^3.

Hope this helps. 

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