The ratio of the radii of two spheres is 1 : 3. Find the ratio of their volume.

Answer

To find the ratio of the volumes of two spheres given the ratio of their radii, we can follow these steps:

### Step 1: Understand the given ratio of the radii
The ratio of the radii of the two spheres is given as \( r_1 : r_2 = 1 : 3 \). This means:
\[ \frac{r_1}{r_2} = \frac{1}{3} \]

### Step 2: 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 3: Write the volumes of the two spheres
Using the formula for volume, we can express the volumes of the two spheres:
- Volume of the first sphere \( V_1 \):
\[ V_1 = \frac{4}{3} \pi r_1^3 \]
- Volume of the second sphere \( V_2 \):
\[ V_2 = \frac{4}{3} \pi r_2^3 \]

### Step 4: Set up the ratio of the volumes
To find the ratio of the volumes \( V_1 : V_2 \), we can write:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} \]

### Step 5: Simplify the ratio
The \( \frac{4}{3} \pi \) terms cancel out:
\[ \frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} \]

### Step 6: Substitute the ratio of the radii
Now we substitute \( \frac{r_1}{r_2} = \frac{1}{3} \):
\[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 = \left(\frac{1}{3}\right)^3 \]

### Step 7: Calculate the cube of the ratio
Calculating \( \left(\frac{1}{3}\right)^3 \):
\[ \left(\frac{1}{3}\right)^3 = \frac{1^3}{3^3} = \frac{1}{27} \]

### Step 8: Write the final ratio of the volumes
Thus, the ratio of the volumes of the two spheres is:
\[ V_1 : V_2 = 1 : 27 \]