Question

12. The ratio of surface area of a sphere and
curved surface area of a hemisphere is 9:
2
then find the ratio of their volumes.​

Answer 1

Solution :-

Let,

Radius of the sphere = R

Radius of the hemisphere = r

Given :-

The ratio of surface area of the sphere and curved surface area of the hemisphere is 9 : 2.

$\bullet \bf \ Surface \ area \ of \ the \ sphere= 4 \pi R^2\\\\\bullet \ Surface \ area \ of \ the \ hemisphere = 2 \pi r^2$

$\longrightarrow \sf \dfrac{4\pi R^2}{2 \pi r^2 }= \dfrac{9}{2} \\\\\longrightarrow \dfrac{2 R^2}{r^2}= \dfrac{9}{2} \\\\\longrightarrow 4R^2 = 9r^2 \\\\\longrightarrow \dfrac{R^2}{r^2} = \dfrac{9}{4} \\\\\longrightarrow \dfrac{R}{r} = \dfrac{3}{2}$

$\bullet \bf \ Volume \ of \ sphere = \dfrac{4}{3} \pi R^3 \\\\\bullet \ Volume \ of \ hemisphere = \dfrac{2}{3} \pi r^3$

$\longrightarrow \sf \dfrac{\dfrac{4}{3} \pi R^3}{\dfrac{2}{3} \pi r^3} \\\\\longrightarrow \dfrac{4\times \pi \times R^3 \times 3 }{ 3 \times 2 \times \pi \times r^3} \\\\\longrightarrow 2 \times \dfrac{R^3}{r^3} \\\\\longrightarrow 2 \times \left( \dfrac{R}{r} \right)^3 \\\\\longrightarrow 2 \times \dfrac{3}{2} \times \dfrac{3}{2} \times \dfrac{3}{2} \\\\\longrightarrow \dfrac{27}{4}$

Ratio of their volumes = 27 : 4