1.

A sphere of maximum volume is cut out from a solid hemisphere of radius r. Find the ratio of the volume of the hemisphere to that of the sphere.1. 4 : 12. 1 : 43. 1 : 54. 2 : 4

Answer» Correct Answer - Option 1 : 4 : 1

Given:

A sphere of maximum volume is cut out from a solid hemisphere of radius r.

Concept used:

Volume of hemisphere = (2/3)πr3

Calculation:

A sphere of maximum volume is cut.

Hence the diameter of sphere = r

Volume = \(\frac{4}{3} \times \pi \times {\left( {\frac{r}{2}} \right)^3}\)

⇒ \(\frac{4}{3} \times \pi \times \frac{{{r^3}}}{8}\)

⇒ \(\frac{{\pi {r^3}}}{6}\)

Required Ratio of the volume of the hemisphere to that of the sphere

⇒ \(\frac{2}{3}\pi {r^3}:\frac{1}{6}\pi {r^3}\)

∴ 4 : 1


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