If the surface areas of the two parts are equal, then the volumes of the two parts are in the ratio:

Question

A solid is hemispherical at the bottom and conical above it. If the surface areas of the two parts are equal, then the volumes of the two parts are in the ratio:

(a) 1 : \(\sqrt{3}\) 

(b) 2: \(\sqrt{3}\)

(c) 3: \(\sqrt{3}\)

(d) 1 : 1 

Answer 1

Answer: (b) \(= 2:\sqrt{3}\)

Let the radius of the hemispherical as well as base of conical portion = r cm

Let the vertical height of the cone = h cm 

Let the slant height of the cone = l cm. 

Given, πrl = 2πr² 

⇒ l = 2 r

\(\therefore \, h = \sqrt{l^2-r^2} = \sqrt{4r^2-r^2}= \sqrt{3r^2} = r\sqrt{3}\)   

∴   Required ratio = Vol. of hemisphere : Vol. of cone 

= \(\frac{2}{3}\pi r^3:\frac{1}{3}\pi r^2.r\sqrt{3}\)  

 \(= 2:\sqrt{3}\)