Respuesta :
Let the radius of the 1st sphere be denoted by [tex] r_1 [/tex]
Let the radius of the 2nd sphere be denoted by [tex] r_2 [/tex]
We know that the surface area of any sphere with radius,r is given by the formula: [tex] S=4\pi r^2 [/tex] where S is the surface area.
Now, let the surface area of the 1st sphere be represented by [tex] S_1 [/tex]. Therefore, [tex] S_1=4\pi r_1^2 [/tex]
Likewise, for the second sphere we will have:[tex] S_2=4\pi r_2^2 [/tex]
Now, we have been given that: [tex] \frac{S_1}{S_2}=\frac{1}{16} [/tex]
[tex] \therefore \frac{4\pi r_1^2}{4\pi r_2^2}=\frac{1}{16} [/tex]
[tex] \therefore \frac{r_1}{r_2}=\frac {1}{16} [/tex]
[tex] \therefore \frac{r_1}{r_2}=\sqrt{\frac{1}{16}}=\frac{1}{4} [/tex]
Now, we know that the Volume, V of any sphere of radius,r is given by the formula: [tex] V=\frac{4}{3}\pi r^3 [/tex]
Thus, the ratio of the volumes of the 1st and the 2nd spheres will be given by:
[tex] \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} [/tex] (where symbols have their usual meanings)
Therefore, [tex] \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} [/tex]
Thus, the ratio of their volumes is [tex] \frac{1}{64} [/tex]
