The surface area of two sphere are in the ratio 49:25. Then there ratio of volume will be

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.

Understand how to find the ratio of volumes for two spheres whose surface areas are in the ratio 49:25. Detailed step-by-step solution covering sphere surface area and volume formulas. Suitable for high school students.

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