Question

The shape of the lower portion of a solid is hemisphere and the shape of upper
portion of it is right circular cone. If the surface areas of two parts are equal, then let
us write by calculating, the ratio of the radius and height of the cone.​

Answer 1

A solid is formed by joining hemisphere and right circular cone

So, let Radius of hemisphere = Base radius of cone = r units

Let the height of the cone be h units

Given :

Surface area of hemisphere = Surface area of right circular cone

Here surface area should be considered as Curved surface area as they are combined

We know that

• Surface area of hemisphere = 2πr² sq.units
• Surface area of cone = πrl

⇒ 2πr² = πrl

Now, Slant height l = √( r² + h² )

⇒ 2πr² = πr × √( r² + h² )

Cancelling πr on both sides since radius are same

⇒ 2r = √( r² + h² )

Squaring on both sides

⇒ ( 2r )² = ( √( r² + h² ) )²

⇒ 4r² = r² + h²

⇒ 4r² - r² = h²

⇒ 3r² = h²

⇒ r² / h² = 1 / 3

⇒ ( r / h )² = 1 / 3

Taking square root on both sides

⇒ √( r / h )² = 1 / √3

⇒ r / h = 1 / √3

⇒ r : h = 1 : √3

Therefore the ratio of radius and height is 1 : √3.

Answer 2

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