If the surface area of two spheres is in the ratio 49 : 25, then the ratio of their volumes will be:

Options

A. `64 : 27`
B. `25: 49`
C. `343 : 64`
D. `343: 125`

Answer

To solve the problem, we need to find the ratio of the volumes of two spheres given the ratio of their surface areas.

### Step 1: Understand the relationship between surface area and radius
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Set up the ratio of surface areas
Given that the surface areas of two spheres are in the ratio \( 49:25 \), we can express this as:
\[ \frac{S_1}{S_2} = \frac{49}{25} \]
Substituting the formula for surface area, we have:
\[ \frac{4\pi r_1^2}{4\pi r_2^2} = \frac{49}{25} \]
The \( 4\pi \) cancels out, simplifying to:
\[ \frac{r_1^2}{r_2^2} = \frac{49}{25} \]

### Step 3: Find the ratio of the radii
Taking the square root of both sides gives:
\[ \frac{r_1}{r_2} = \frac{\sqrt{49}}{\sqrt{25}} = \frac{7}{5} \]

### Step 4: Use the ratio of the radii to find the ratio of volumes
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]
Thus, the ratio of the volumes of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} \]
Again, the \( \frac{4}{3}\pi \) cancels out, leading to:
\[ \frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} \]

### Step 5: Substitute the ratio of the radii
Now substituting the ratio of the radii:
\[ \frac{V_1}{V_2} = \frac{\left(\frac{7}{5}\right)^3}{1} = \frac{7^3}{5^3} = \frac{343}{125} \]

### Conclusion
Thus, the ratio of the volumes of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{343}{125} \]

### Final Answer
The ratio of their volumes is \( 343:125 \).

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