Question

The ratio of volumes of two spheres is 512:729. find the ratio of their surface areas.

Answer 1

Answer:

${A_1}:{A_2}=64:81$

Step-by-step explanation:

Formula used:

Volume of a sphere of radius r

$=\frac{4}{3}\pi\:r^3$ cubic units

Surface area of a sphere of radius r

$=4\pi\:r^2$ square units

Let $r_1\:and\:r_2$ be radii of the given two spheres.

Given:

$V_1:V_2=512:729$

$\frac{4}{3}\pi\:{r_1}^3:\frac{4}{3}\pi\:{r_2}^3=512:729$

$\frac{\frac{4}{3}\pi\:{r_1}^3}{\frac{4}{3}\pi\:{r_2}^3}=\frac{512}{729}$

$\frac{{r_1}^3}{{r_2}^3}=\frac{8^3}{9^3}$

$(\frac{r_1}{r_2})^3=(\frac{8}{9})^3$

this implies

$\frac{r_1}{r_2}=\frac{8}{9}$

$r_1=8k\:and\:r_2=9k$

Now,

$\frac{A_1}{A_2}$

$=\frac{4\pi\:{r_1}^2}{4\pi\:{r_2}^2}$

$=\frac{{r_1}^2}{{r_2}^2}$

$=\frac{(8k)^2}{(9k)^2}$

$=\frac{64k^2}{81k^2}$

$=\frac{64}{81}$

${A_1}:{A_2}=64:81$