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

Answer

To solve the problem, we need to find the ratio of the surface areas of two spheres given that the volume of one sphere is 27 times that of the other.

### Step-by-Step Solution:

1. **Understand the Relationship Between Volumes**:
We know that the volume of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Let the volume of the first sphere be \( V_1 \) and the volume of the second sphere be \( V_2 \). According to the problem:
\[ V_1 = 27 V_2 \]

2. **Express Volumes in Terms of Radii**:
Substitute the volume formulas into the equation:
\[ \frac{4}{3} \pi r_1^3 = 27 \left( \frac{4}{3} \pi r_2^3 \right) \]
We can cancel \( \frac{4}{3} \pi \) from both sides:
\[ r_1^3 = 27 r_2^3 \]

3. **Find the Ratio of Radii**:
To find the ratio of the radii, we take the cube root of both sides:
\[ \frac{r_1}{r_2} = \sqrt[3]{27} = 3 \]
Thus, we have:
\[ r_1 : r_2 = 3 : 1 \]

4. **Calculate the Surface Areas**:
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Now, we need to find the ratio of the surface areas of the two spheres:
\[ \frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} \]
This simplifies to:
\[ \frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} \]

5. **Substitute the Ratio of Radii**:
We already found that \( \frac{r_1}{r_2} = 3 \). Therefore:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = 3^2 = 9 \]
So, the ratio of the surface areas is:
\[ A_1 : A_2 = 9 : 1 \]

### Final Answer:
The ratio of the surface areas of the two spheres is \( 9 : 1 \).