Question

The surface area of two spheres are in a ratio of 1:16. What is the ratio of their volume?

Answer 1

Let the radius of the 1st sphere be denoted by $r_1$

Let the radius of the 2nd sphere be denoted by $r_2$

We know that the surface area of any sphere with radius,r is given by the formula: $S=4\pi r^2$ where S is the surface area.

Now, let the surface area of the 1st sphere be represented by $S_1$. Therefore, $S_1=4\pi r_1^2$

Likewise, for the second sphere we will have:$S_2=4\pi r_2^2$

Now, we have been given that: $\frac{S_1}{S_2}=\frac{1}{16}$

$\therefore \frac{4\pi r_1^2}{4\pi r_2^2}=\frac{1}{16}$

$\therefore \frac{r_1}{r_2}=\frac {1}{16}$

$\therefore \frac{r_1}{r_2}=\sqrt{\frac{1}{16}}=\frac{1}{4}$

Now, we know that the Volume, V of any sphere of radius,r is given by the formula: $V=\frac{4}{3}\pi r^3$

Thus, the ratio of the volumes of the 1st and the 2nd spheres will be given by:

$\frac{V_1}{V_2}=\frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} =\frac{r_1^3}{r_2^3}$ (where symbols have their usual meanings)

Therefore, $\frac{V_1}{V_2}=\frac{r_1^3}{r_2^3}=\frac{r_1^2}{r_2^2}\times \frac{r_1}{r_2}=(\frac{r_1}{r_2})^2\times \frac{r_1}{r_2}=\frac{1}{16}\times \frac{1}{4}=\frac{1}{64}$

Thus, the ratio of their volumes is $\frac{1}{64}$

Answer 2

3.141592653589793238463642 which is few of pi