The ratio of the surfaces of two spheres is 2 : 3. Find the ratio of their volumes.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the surface areas of the spheres.
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 the surface areas.
We are given that the ratio of the surface areas of two spheres is \( 2:3 \). Therefore, we can write:
\[ \frac{S_1}{S_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{r_1^2}{r_2^2} = \frac{2}{3} \]

### Step 3: Express the ratio of the radii.
From the equation \( \frac{r_1^2}{r_2^2} = \frac{2}{3} \), we can take the square root of both sides to find the ratio of the radii:
\[ \frac{r_1}{r_2} = \sqrt{\frac{2}{3}} = \frac{\sqrt{2}}{\sqrt{3}} \]

### Step 4: Write the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 5: Set up the ratio of the volumes.
Now we can find the ratio of the volumes of the two spheres:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{r_1^3}{r_2^3} \]

### Step 6: Substitute the ratio of the radii into the volume ratio.
We already found \( \frac{r_1}{r_2} = \frac{\sqrt{2}}{\sqrt{3}} \). Therefore,
\[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 = \left(\frac{\sqrt{2}}{\sqrt{3}}\right)^3 = \frac{(\sqrt{2})^3}{(\sqrt{3})^3} = \frac{2\sqrt{2}}{3\sqrt{3}} \]

### Step 7: Finalize the ratio of the volumes.
Thus, the ratio of the volumes of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{2\sqrt{2}}{3\sqrt{3}} \]

### Summary
The ratio of the volumes of the two spheres is \( \frac{2\sqrt{2}}{3\sqrt{3}} \).

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