Question

· The surface areas of two spheres are in the ratio 1: 4. Find the ratio of
their volumes.

Answer 1

Given :

• The surface areas of two spheres are in the ratio 1: 4.

To find :

• The ratio of their volumes =?

Step-by-step explanation :

We know that,

Area of sphere = 4πr²

And

Volume of sphere = (4/3)πr³

Now,

Let the radii of two spheres be r and R respectively.

According to the question :

➮ (4 × π × r²)/(4 × π × R²) = 1/4

➮ (r/R)² = (1/2)²

➮ r/R = 1/2

As We know that,

Volume of sphere = (4/3)πr³

Now, the ratio of their volumes is

= ((4/3) × π × r³)/((4/3)× π × R³)

= (r/R)³

On putting the value of r/R = 1/2 we get,

= (1/2)³

= 1/8

Therefore, the ratio of their volumes = 1:8

Answer 2

$\purple{\underline{\underline{\pink{\boxed{\boxed{\red{\star{\sf Question:}}}}}}}}$

$\rm{The \ surface \ areas \ of \ two \ spheres \ are \ in}$

$\rm{the \ ratio \ 1:4. \ Find \ the \ ratio \ of \ their}$

$\rm{volumes}$

_________________________________________

$\purple{\underline{\underline{\orange{\boxed{\boxed{\green{\star{\sf Answer:}}}}}}}}$

$\rm{\blue{\boxed{\red{ ratio \ of \ volume \ of \ two \ spheres \ is \ 1 : 8}}}}$

_________________________________________

$\sf\large\underline\pink{Given:}$

$\rm{The \ surface \ areas \ of \ two \ spheres }$

$\rm{are \ in \ the \ ratio \ 1:4.}$

_________________________________________

$\sf\large\underline\blue{To \ Find}$

$\rm{ratio \ of \ volume \ of \ two \ spheres }$

_________________________________________

$\purple{\underline{\underline{\red{\boxed{\boxed{\blue{\star{\sf Solution:}}}}}}}}$

$\rm{We \ are \ given ,}$

$\rm{The \ surface \ areas \ of \ two \ spheres }$

$\rm{are \ in \ the \ ratio \ 1:4.}$

$\rm{Let \ r_1 \ and \ r_2 \ be \ the \ raddii \ of }$

$\rm{the \ two \ spheres}$

$\rm\implies{r_1:r_2 \ = \ 1:4}$

$\rm{Now, \ surface \ area \ of \ the \ spheres }$

$\rm{are \ 4 \pi r_1 \ and \ 4 \pi r_2}$

$\rm\implies{ {\frac{4 \pi r_1}{4 \pi r_2} }^{2} \ = \ \frac{1}{4}}$

$\rm\implies{\frac{{r_1}^2}{{r_2}^{2}} \ = \ \frac{1}{4}}$

$\rm\implies{\frac{r_1}{r_2} \ = \ \sqrt{ \frac{1}{4}} }$

$\rm\implies{\red{\boxed{\green{ \frac{r_1}{r_2} \ = \ \frac{1}{2}} }}}$

$\rm{Now, \ volume \ of \ the \ spheres \ are}$

$\rm\implies{( \frac{4}{3}) \pi {r_1}^{3} \ and ( \frac{4}{3}) \pi {r_2}^{3}}$

$\rm\implies{Ratio \ of \ volumes : \ \frac{ ( \frac{4}{3}) \pi {r_1}^{3}}{ (\frac{4}{3}) \pi {r_1}^{3}}}$

$\rm\implies{Ratio \ of \ volumes : \ \frac{{r_1}^{3}}{{r_2}^{3}}}$

$\rm\implies{Ratio \ of \ volumes : \ {( \frac{ r_1}{ r_2} ) }^{3}}$

$\rm\implies{Ratio \ of \ volumes : \ { ( \frac{ 1}{ 2}) }^{3} }$

$\rm\implies{Ratio \ of \ volumes : \ { \frac{1}{8}}}$

$\rm\therefore{\blue{\boxed{\red{ ratio \ of \ volume \ of \ two \ spheres \ is \ 1 : 8}}}}$

_________________________________________

$\rm\orange{Hope \ it \ helps \ :))}$