Excess pressure inside 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. `1:4`
C. `64:1`
D. `1:2`

Explanation

To solve the problem of finding the ratio of the volumes of two soap bubbles given that the excess pressure inside one bubble is four times that of the other, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between pressure and radius**:
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 and \( R \) is the radius of the bubble.

2. **Set up the equations for the two bubbles**:
Let the excess pressure of the first bubble be \( P_1 \) and that of the second bubble be \( P_2 \). According to the problem, we have:
\[ P_1 = 4P_2 \]

3. **Express the pressures in terms of radii**:
Using the formula for excess pressure, we can write:
\[ P_1 = \frac{4T}{R_1} \quad \text{and} \quad P_2 = \frac{4T}{R_2} \]

4. **Substitute the pressures into the equation**:
From the relationship \( P_1 = 4P_2 \), we substitute:
\[ \frac{4T}{R_1} = 4 \left(\frac{4T}{R_2}\right) \]

5. **Simplify the equation**:
Cancel \( 4T \) from both sides (assuming \( T \neq 0 \)):
\[ \frac{1}{R_1} = \frac{16}{R_2} \]

6. **Cross-multiply to find the ratio of radii**:
This gives us:
\[ R_2 = 16R_1 \]

7. **Find the ratio of volumes**:
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the volumes of the two bubbles can be expressed as:
\[ V_1 = \frac{4}{3} \pi R_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi R_2^3 \]
The ratio of the volumes is:
\[ \frac{V_1}{V_2} = \frac{R_1^3}{R_2^3} \]

8. **Substituting the ratio of radii**:
Since \( R_2 = 16R_1 \), we have:
\[ \frac{V_1}{V_2} = \frac{R_1^3}{(16R_1)^3} = \frac{R_1^3}{16^3 R_1^3} = \frac{1}{16^3} \]

9. **Calculate \( 16^3 \)**:
\[ 16^3 = 4096 \]
Thus, the ratio of the volumes is:
\[ \frac{V_1}{V_2} = \frac{1}{4096} \]

10. **Final ratio**:
Therefore, the ratio of the volume of the first bubble to the second bubble is:
\[ V_1 : V_2 = 1 : 4096 \]

### Final Answer:
The ratio of the volume of the first bubble to the second bubble is \( 1 : 4096 \).