If the radius of a soap bubble is four times that of another, then the ratio of their pressures will be

Options

A. `1:4`
B. `4:1`
C. `16:1`
D. `1:16`

Explanation

To solve the problem, we need to find the ratio of the pressures of two soap bubbles, where the radius of one bubble is four times that of the other. Let's denote the radius of the smaller bubble as \( r \) and the radius of the larger bubble as \( R = 4r \).

### Step-by-step Solution:

1. **Understand the relationship between pressure and radius:**
The pressure inside a soap bubble is given by the formula:
\[ P = \frac{4S}{r} \]
where \( S \) is the surface tension of the soap solution and \( r \) is the radius of the bubble.

2. **Write the pressure for both bubbles:**
- For the smaller bubble (radius \( r \)):
\[ P_1 = \frac{4S}{r} \]
- For the larger bubble (radius \( R = 4r \)):
\[ P_2 = \frac{4S}{R} = \frac{4S}{4r} = \frac{S}{r} \]

3. **Find the ratio of the pressures:**
Now, we can find the ratio of the pressures \( P_1 \) and \( P_2 \):
\[ \frac{P_1}{P_2} = \frac{\frac{4S}{r}}{\frac{S}{r}} = \frac{4S}{r} \cdot \frac{r}{S} = 4 \]

4. **Conclusion:**
The ratio of the pressures of the two soap bubbles is:
\[ \frac{P_1}{P_2} = 4:1 \]

### Final Answer:
The ratio of the pressures of the two soap bubbles is \( 4:1 \).