Given that the ratio of the surface areas of two spheres is 49:25, we are asked to find the ratio of their volumes.

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

Step 1: Relating Surface Area and Radius

The surface area $A$ of a sphere is given by: $A = 4 \pi r^2$

Since the ratio of the surface areas is 49:25, we have: $\frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{r_1^2}{r_2^2} = \frac{49}{25}$

Taking the square root of both sides to relate the radii: $\frac{r_1}{r_2} = \frac{\sqrt{49}}{\sqrt{25}} = \frac{7}{5}$

Step 2: Relating Volume and Radius

The volume $V$ of a sphere is given by: $V = \frac{4}{3} \pi r^3$

The ratio of the volumes of the two spheres is: $\frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{r_1^3}{r_2^3}$

Using the ratio of the radii $\frac{r_1}{r_2} = \frac{7}{5}$, we can find the ratio of the volumes: $\frac{V_1}{V_2} = \left(\frac{7}{5}\right)^3 = \frac{343}{125}$

Final Answer:

The ratio of the volumes of the two spheres is $343:125$.

Would you like further details or have any questions?

Here are five related questions to consider:

1. How do surface area and volume formulas change for other 3D shapes?
2. What is the effect on volume if the radius of a sphere doubles?
3. Can you derive the formulas for surface area and volume of a sphere?
4. How would you solve this problem if given the diameter instead of the radius?
5. What is the relationship between surface area and volume ratios for any geometric solids?

Tip: Always remember that volume scales with the cube of the radius, while surface area scales with the square.