Two soap bubbles in vacuum of radius `3cm` and` 4cm` coalesce to form a single bubble, in same temperature. What will be the radius of the new bubble

Options

A. `7cm`
B. `(12)/(7) cm`
C. `12cm`
D. `5cm`

Explanation

To find the radius of the new soap bubble formed by the coalescence of two soap bubbles with radii 3 cm and 4 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two soap bubbles in a vacuum with radii \( r_1 = 3 \, \text{cm} \) and \( r_2 = 4 \, \text{cm} \). We need to find the radius \( R \) of the new bubble formed when these two bubbles coalesce.

2. **Use the Concept of Volume**: When two bubbles coalesce, the total volume of the new bubble is equal to the sum of the volumes of the two original bubbles. The volume \( V \) of a sphere (bubble) is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

3. **Calculate the Volumes of the Original Bubbles**:
- For the first bubble (radius \( r_1 = 3 \, \text{cm} \)):
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \, \text{cm}^3 \]
- For the second bubble (radius \( r_2 = 4 \, \text{cm} \)):
\[ V_2 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3 \]

4. **Sum the Volumes**:
\[ V_{total} = V_1 + V_2 = 36 \pi + \frac{256}{3} \pi \]
To add these, convert \( 36 \) to a fraction:
\[ 36 = \frac{108}{3} \]
So,
\[ V_{total} = \frac{108}{3} \pi + \frac{256}{3} \pi = \frac{364}{3} \pi \, \text{cm}^3 \]

5. **Find the Radius of the New Bubble**: Let the radius of the new bubble be \( R \). The volume of the new bubble is:
\[ V_{new} = \frac{4}{3} \pi R^3 \]
Setting this equal to the total volume:
\[ \frac{4}{3} \pi R^3 = \frac{364}{3} \pi \]

6. **Solve for \( R^3 \)**:
\[ R^3 = \frac{364}{4} = 91 \]

7. **Calculate \( R \)**:
\[ R = \sqrt[3]{91} \]
Approximating \( \sqrt[3]{91} \):
\[ R \approx 4.48 \, \text{cm} \]

### Final Answer:
The radius of the new bubble is approximately \( 4.48 \, \text{cm} \).