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A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surface will be :

A. $$\sqrt 2 :1$$

B. $$1:\sqrt 2 $$

C. 2 : 1

D. 1 : 2

Answer: Option A

Solution(By Examveda Team)

Volume and Surface Area mcq solution image
Let,
$$\eqalign{ & OP = OQ = OR = r \cr & \therefore OR = h = r \cr} $$
∴ Curved surface area of the hemisphere = $$2\pi {r^2}$$
Curved surface area of a cone = $$\pi rl$$
Where,
$$\eqalign{ & l = \sqrt {{h^2} + {r^2}} \cr & \,\,\,\,\, = \sqrt {{r^2} + {r^2}} \cr & \,\,\,\,\, = r\sqrt 2 \cr} $$
∴ Required ratio :
$$\eqalign{ & = \frac{{2\pi {r^2}}}{{\pi rl}} \cr & = \frac{{2\pi {r^2}}}{{\pi r \times r\sqrt 2 }} \cr & = \frac{2}{{\sqrt 2 }} \cr & = \frac{{2 \times \sqrt 2 }}{{\sqrt 2 \times \sqrt 2 }} \cr & = \frac{{2\sqrt 2 }}{2} \cr & = \frac{{\sqrt 2 }}{1}\,Or\,\sqrt 2 :1 \cr} $$

This Question Belongs to Arithmetic Ability >> Volume And Surface Area

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