A soap bubble in vacuum has a radius of 3 cm ad another soap bubble in vacuum has a radius of 4 cm. if the two bubbles coalesce under isothermal condition, then the radius of the new bubble is

Options

A. 2.3 cm
B. 4.5 cm
C. 5 cm
D. 7 cm

Answer

To solve the problem of finding the radius of the new soap bubble formed when two soap bubbles coalesce, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two soap bubbles in a vacuum with radii \( r_1 = 3 \) cm and \( r_2 = 4 \) cm. When they coalesce, they form a new bubble. We need to find the radius \( R \) of this new bubble.

2. **Use the Formula for Volume**: The volume of a soap bubble is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Therefore, the volumes of the two bubbles are:
- For bubble 1 (radius \( r_1 \)):
\[ V_1 = \frac{4}{3} \pi (r_1)^3 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi \text{ cm}^3 \]
- For bubble 2 (radius \( r_2 \)):
\[ V_2 = \frac{4}{3} \pi (r_2)^3 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

3. **Calculate the Total Volume**: When the two bubbles coalesce, the total volume \( V \) of the new bubble is the sum of the volumes of the two bubbles:
\[ V = V_1 + V_2 = 36\pi + \frac{256}{3}\pi \]
To add these, we need a common denominator:
\[ V = \frac{108}{3}\pi + \frac{256}{3}\pi = \frac{364}{3}\pi \text{ cm}^3 \]

4. **Set the Volume of the New Bubble**: The volume of the new bubble with radius \( R \) is:
\[ V = \frac{4}{3} \pi R^3 \]
Setting this equal to the total volume we found:
\[ \frac{4}{3} \pi R^3 = \frac{364}{3} \pi \]

5. **Solve for \( R^3 \)**: Cancel \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = \frac{364}{4} = 91 \]

6. **Find \( R \)**: Take the cube root of both sides to find \( R \):
\[ R = \sqrt[3]{91} \approx 4.48 \text{ cm} \]

7. **Final Answer**: The radius of the new bubble is approximately \( 4.48 \) cm.