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ML Aggarwal Solutions Class 10 Mathematics Solutions for Mensuration Exercise 17.2 in Chapter 17 - Mensuration
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The height and the radius of the base of a right circular cone is half the corresponding height and radius of another bigger cone.
Find: the ratio of their lateral surface areas.
Solution:
The shape of a cone is produced by stacking several triangles and spinning them around an axis. It has a total surface area and a curved surface area because it has a flat base.
Let r be the radius of bigger cone. Then the radius of smaller cone is r/2.
Let h be the height of bigger cone. Then the height of smaller cone is h/2.
slant height of bigger cone = √(h2+r2)
slant height of smaller cone = √((h/2)2+(r/2)2) = √(h2/4+r2/4) = ½ √(h2+r2)
Curved surface area of bigger cone, s1 = π rl
=π r√(h2+r2)
Curved surface area of smaller cone, s2 =π rl
= π ×(r/2)× ½ √(h2+r2)
= ¼ π r√(h2+r2)
ratio of curved surface area of smaller cone to bigger cone, s2/s1 = ¼ π r√(h2+r2) ÷ π r√(h2+r2)
= ¼ r√(h2+r2) ×1/(r√(h2+r2))
= 1/4
Hence the ratio of curved surface area of smaller cone to bigger cone is 1:4 
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