The volumes of two spheres are in the ratio of 512: 3375. The ratio of their surface areas is:

The correct option is 64 : 225.

Step-by-step Explanation:

1. Formula for the Volume of a Sphere:

The volume $V$ of a sphere with radius $r$ is given by the formula:

$V=\frac{4}{3}\pi r^3$

2. Finding the Ratio of their Radii:

Let $r_1$ and $r_2$ be the radii of the two spheres, and $V_1$ and $V_2$ be their respective volumes.

We are given that the ratio of their volumes is:

$\frac{V_1}{V_2}=\frac{512}{3375}$

Substituting the volume formula for both spheres:

$\frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3}=\frac{512}{3375}$

Canceling out the common term $\frac{4}{3}\pi$ from the numerator and denominator:

${\left(\frac{r_1}{r_2}\right)}^3=\frac{512}{3375}$

Taking the cube root on both sides:

$\frac{r_1}{r_2}=\sqrt[3]{\frac{512}{3375}}$

Since $8^3=512$ and ${15}^3=3375$, we get:

$\frac{r_1}{r_2}=\frac{8}{15}$

3. Calculating the Ratio of their Surface Areas:

The surface area $S$ of a sphere with radius $r$ is given by:

$S=4\pi r^2$

Let $S_1$ and $S_2$ be the surface areas of the two spheres. Their ratio is:

$\frac{S_1}{S_2}=\frac{4\pi r_1^2}{4\pi r_2^2}$

Canceling out $4\pi$:

$\frac{S_1}{S_2}={\left(\frac{r_1}{r_2}\right)}^2$

Substitute $\frac{r_1}{r_2}=\frac{8}{15}$ into the equation:

$\frac{S_1}{S_2}={\left(\frac{8}{15}\right)}^2=\frac{64}{225}$

Thus, the ratio of their surface areas is 64 : 225.