The excess pressure inside one soap bubble is three times that inside a second soap bubble, then the ratio of their surface area is:

Options

A. `1 : 9`
B. `1 : 3`
C. `3 : 1`
D. `1 : 27`

Answer

To solve the problem, we need to understand the relationship between the excess pressure inside soap bubbles and their radii, and subsequently how this relates to their surface areas.

### Step-by-Step Solution:

1. **Understanding Excess Pressure in Soap Bubbles**:
The excess pressure (ΔP) inside a soap bubble is given by the formula:
\[ \Delta P = \frac{4T}{R} \]
where \(T\) is the surface tension of the soap solution and \(R\) is the radius of the bubble.

2. **Setting Up the Relationship**:
Let’s denote the excess pressure inside the first soap bubble as \(P_1\) and the second soap bubble as \(P_2\). According to the problem, we have:
\[ P_1 = 3P_2 \]

3. **Expressing the Pressures in Terms of Radii**:
Using the formula for excess pressure:
\[ P_1 = \frac{4T}{R_1} \quad \text{and} \quad P_2 = \frac{4T}{R_2} \]
Substituting these into the relationship gives:
\[ \frac{4T}{R_1} = 3 \cdot \frac{4T}{R_2} \]

4. **Simplifying the Equation**:
We can cancel \(4T\) from both sides (assuming \(T \neq 0\)):
\[ \frac{1}{R_1} = \frac{3}{R_2} \]
Rearranging this gives:
\[ R_2 = 3R_1 \]

5. **Finding the Ratio of Surface Areas**:
The surface area \(S\) of a soap bubble is given by:
\[ S = 4\pi R^2 \]
Therefore, the surface areas of the two bubbles can be expressed as:
\[ S_1 = 4\pi R_1^2 \quad \text{and} \quad S_2 = 4\pi R_2^2 \]
The ratio of their surface areas is:
\[ \frac{S_1}{S_2} = \frac{4\pi R_1^2}{4\pi R_2^2} = \frac{R_1^2}{R_2^2} \]

6. **Substituting the Radius Relationship**:
Since we found \(R_2 = 3R_1\), we can substitute this into the ratio:
\[ \frac{S_1}{S_2} = \frac{R_1^2}{(3R_1)^2} = \frac{R_1^2}{9R_1^2} = \frac{1}{9} \]

7. **Final Ratio of Surface Areas**:
Thus, the ratio of the surface areas of the two soap bubbles is:
\[ \frac{S_1}{S_2} = \frac{1}{9} \]

### Conclusion:
The ratio of the surface areas of the two soap bubbles is \(1:9\).