Question

the surface area of two spheres are in ratio 1:2 . find the ratio of their volumes

Answer 1

surface area of sphere is 4πr^2.
let the two radii be R1 and R2
•4π(R1)^2/4π(R2)^2= 1/2
this gives (R1)^2/(R2)^2=1/2
R1/R2=1/√2
volume of sphere =4/3πr^3
ratio of their volume gives
(R1)^3/(R2)^3=R1×(R1)^2/R2×(R2)^3
=1×1/2×√2
=1/2√2

Answer 2

The ratio of their volumes $=1:2\sqrt{2}$.

Step-by-step explanation:

Let the radius of two spheres =  r and R

Given,

The ratio of the surface area of two spheres = 1 : 2

To find, the ratio of their volumes = ?

We know that,

The surface area of sphere $=4\pi r^{2}$

⇒ $\dfrac{4\pi r^{2}}{4\pi R^{2}}=\dfrac{1}{2}$

⇒ $\dfrac{r^{2}}{R^{2}}=\dfrac{1}{2}=(\dfrac{1}{\sqrt{2}})^2$

⇒ $\dfrac{r}{R}=\dfrac{1}{\sqrt{2}}$

The ratio of the volume of two spheres,

$\dfrac{\dfrac{4}{3}\pi r^{3}}{\dfrac{4}{3} \pi R^{3}}$

$=\dfrac{r^{3}}{R^{3}}$

$(\dfrac{r}{R} )^{3} =(\dfrac{1}{\sqrt{2}} )^{3}$

$=\dfrac{1}{2\sqrt{2}}$

$=1:2\sqrt{2}$

Hence, the ratio of their volumes $=1:2\sqrt{2}$.