The volume of one sphere is 27 times that of another sphere. Calculate the ratio of their :
radii,

Answer

To solve the problem of finding the ratio of the radii of two spheres where the volume of one sphere is 27 times that of the other, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Define the Radii**:
Let the radius of the larger sphere be \( R \) and the radius of the smaller sphere be \( r \).

3. **Write the Volume Equations**:
- Volume of the larger sphere:
\[ V_1 = \frac{4}{3} \pi R^3 \]
- Volume of the smaller sphere:
\[ V_2 = \frac{4}{3} \pi r^3 \]

4. **Set Up the Volume Ratio**:
According to the problem, the volume of the larger sphere is 27 times that of the smaller sphere:
\[ V_1 = 27 V_2 \]
Substituting the volume formulas:
\[ \frac{4}{3} \pi R^3 = 27 \left( \frac{4}{3} \pi r^3 \right) \]

5. **Cancel Common Terms**:
We can cancel \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 27 r^3 \]

6. **Solve for the Ratio of Radii**:
To find the ratio of the radii, we take the cube root of both sides:
\[ \frac{R}{r} = \sqrt[3]{27} \]

7. **Calculate the Cube Root**:
The cube root of 27 is 3:
\[ \frac{R}{r} = 3 \]

8. **Express the Ratio**:
Therefore, the ratio of the radii of the larger sphere to the smaller sphere is:
\[ R : r = 3 : 1 \]

### Final Answer:
The ratio of the radii of the two spheres is \( 3 : 1 \).

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