Excess pressure one soap bubble is four times that of other. Then the ratio of volume of first bubble to second one is

Options

A. `1:64`
B. `64:1`
C. `4:1`
D. `1:2`

Explanation

To solve the problem, we need to find the ratio of the volumes of two soap bubbles given that the excess pressure inside one bubble is four times that of the other.

### Step-by-Step Solution:

1. **Understanding Excess Pressure in Soap Bubbles**:
The excess pressure (\( \Delta P \)) 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 the excess pressure of the first bubble be \( \Delta P_1 \) and that of the second bubble be \( \Delta P_2 \). According to the problem:
\[ \Delta P_1 = 4 \Delta P_2 \]

3. **Expressing Excess Pressure in Terms of Radius**:
Using the formula for excess pressure, we can write:
\[ \Delta P_1 = \frac{4T}{R_1} \quad \text{and} \quad \Delta P_2 = \frac{4T}{R_2} \]
Substituting these into the relationship gives:
\[ \frac{4T}{R_1} = 4 \left(\frac{4T}{R_2}\right) \]

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

5. **Finding the Volume Ratio**:
The volume \( V \) of a sphere (or bubble) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the volumes of the two bubbles are:
\[ V_1 = \frac{4}{3} \pi R_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi R_2^3 \]
The ratio of the volumes is:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi R_1^3}{\frac{4}{3} \pi R_2^3} = \frac{R_1^3}{R_2^3} \]

6. **Substituting the Radius Relationship**:
Since \( R_2 = 4R_1 \), we have:
\[ \frac{V_1}{V_2} = \frac{R_1^3}{(4R_1)^3} = \frac{R_1^3}{64R_1^3} = \frac{1}{64} \]

7. **Final Result**:
Thus, the ratio of the volumes of the first bubble to the second bubble is:
\[ \frac{V_1}{V_2} = \frac{1}{64} \]

### Conclusion:
The ratio of the volume of the first bubble to the second bubble is \( 1:64 \).