Question

the ratio of surface area of a sphere and curved surface area of a hemisphere is 9:2 find ratio of their volumes

Answer 1

Answer:

Step-by-step explanation:

Consider the radius of the sphere = $r_{1}$

therefore, the total surface area of a sphere is $A_{s}=4\pi\times r_{1}^{2}$

Consider the radius of the hemisphere = $r_{2}$

Curved Surface Area of Hemisphere $A_{hs}=2\pi\times{r_{2}}^{2}$

As given

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

$\therefoer \frac{A_{s}}{A_{hs}}=\frac{4\pi\times r_{1}^{2}}{2\pi\times r_{2}^{2}}\\\\\Rightarrow\frac{9}{2}=2\times\frac{r_{1}^{2}}{r_{2}^{2}}\\\\\Rightarrow\frac{r_{1}^{2}}{r_{2}^{2}}=\frac{9}{4}\\\\\Rightarrow{r_{1}^{2}}=\frac{9}{4}\times{r_{2}^{2}}\\\\\Rightarrow{r_{1}}=\sqrt{\frac{9}{4}\times{r_{2}^{2}}}\\\\\Rightarrow{r_{1}}={\frac{3}{2}\times{r_{2}}\text{ ...............equation-1}$

Volume of the sphere

$V_{s}=\frac{4}{3}\pi\times r_{1}^3\\\\\Rightarrow V_{s}=\frac{4}{3}\pi\times ({\frac{3}{2}\times{r_{2}})^{3}\text{ [put the value of}\text{ }r_{1}}]\\\\\Rightarrow V_{s}=\frac{4}{3}\pi\times\frac{9}{4}\times r_{2}^{3}\\\\\Rightarrow V_{s}=3\pi\times r_{2}^{3}$

Volume of the hemisphere $V_{hs}=\frac{2}{3}\pi\times r_{2}^{3}$

The ratio of their volumes

$\frac{V_{s}}{V_{hs}}=\frac{3\pi\times r_{2}^{3}}{\frac{2}{3}\pi\times r_{2}^{3}}=\frac{9}{2}$