The ratio between the volume of two spheres is 8 : 27.

Question:

The ratio between the volume of two spheres is 8 : 27. What is the ratio between their surface areas?
(a) 2 : 3
(b) 4 : 5
(c) 5 : 6
(d) 4 : 9

Solution:

(d) 4 : 9
Let the radii of the spheres be R and r, respectively.

Then, ratio of their volumes $=\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi \mathrm{r}^{3}}$

Therefore,

$\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}=\frac{8}{27}$

$\Rightarrow \frac{R^{3}}{r^{3}}=\frac{8}{27}$

$\Rightarrow\left(\frac{R}{r}\right)^{3}=\left(\frac{2}{3}\right)^{3}$

$\Rightarrow \frac{R}{r}=\frac{2}{3}$

Hence, the ratio between their surface areas $=\frac{4 \pi R^{2}}{4 \pi r^{2}}$

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

$=\left(\frac{R}{r}\right)^{2}$

$=\left(\frac{2}{3}\right)^{2}$

$=\frac{4}{9}$

$=4: 9$

 

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