Question

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

Answer 1

Answer:

Ratio of Volume of Sphere to Volume of Hemisphere is 27 : 4

Step-by-step explanation:

Given:

Ratio of Surface area of Sphere to Curved surface area of hemisphere = 9 : 2

To find: Ratio of their volumes

Formula of Surface of sphere and volume is given by,

$Surface\,Area\,=\,4\pi r^2$

$Volume\,=\,\frac{4}{3}\times\pi r^3$

And Curved Surface area & Volume of Hemisphere is given by,

$Curved\,Surface\,Area\,=\,2\pi R^2$

$Volume\,=\,\frac{2}{3}\times\pi R^3$

Thus,

$\frac{Surface\,Area\,of\,Sphere}{Curved\,Surface\,Area\,of\,Hemisphere}=\frac{9}{2}$

$\frac{4\pi r^2}{2\pi R^2}=\frac{9}{2}$

$\frac{2\times r^2}{R^2}=\frac{9}{2}$

$\frac{r^2}{R^2}=\frac{9}{4}$

$(\frac{r}{R})^2=\frac{9}{4}$

$\frac{r}{R}=\sqrt{\frac{9}{4}}$

$\frac{r}{R}=\frac{3}{2}$  

(∵ radii are positive value ⇒ only positive value of root is taken)

Now,

$\frac{Volume\,of\,Sphere}{Volume\,of\,Hemisphere}$

$\implies\frac{\frac{4}{3}\times\pi r^3}{\frac{2}{3}\times\pi R^3}$

$\implies\frac{2\times r^3}{R^3}$

$\implies2\times(\frac{r}{R})^3$

$\implies2\times(\frac{3}{2})^3$

$\implies2\times\frac{27}{8}$

$\implies\frac{27}{4}$

Therefore, Ratio of Volume of Sphere to Volume of Hemisphere is 27 : 4