To find the ratio of the volumes of two spheres given the ratio of their surface areas, we can use the formulas for surface area and volume of a sphere.

Let's denote the radii of the two spheres as r_1 and r_2.

The formula for the surface area of a sphere is given by:

A = 4\pi r^2

Therefore, the ratio of the surface areas of the two spheres is:

\frac{4\pi r_1^2}{4\pi r_2^2} = \frac{r_1^2}{r_2^2} = \frac{9}{49}

This implies:

\left(\frac{r_1}{r_2}\right)^2 = \frac{9}{49}

Taking the square root of both sides, we get:

\frac{r_1}{r_2} = \frac{3}{7}

Now, the formula for the volume of a sphere is:

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

So, the ratio of the volumes of the two spheres is:

\frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \frac{r_1^3}{r_2^3}

Substituting the known ratio of the radii, we have:

\frac{r_1^3}{r_2^3} = \left(\frac{3}{7}\right)^3 = \frac{27}{343}

Therefore, the ratio of the volumes of the two spheres is 27:343.