Step 1: Understanding the Concept:
A soap bubble has two surfaces (inner and outer) in contact with air.
Due to surface tension, there is an excess pressure inside the bubble compared to the outside pressure.
This excess pressure is inversely proportional to the radius of the bubble.
Step 2: Key Formula or Approach:
The excess pressure \( P \) inside a soap bubble of radius \( r \) and surface tension \( T \) is given by:
\[ P = \frac{4T}{r} \]
The volume \( V \) of a spherical bubble of radius \( r \) is:
\[ V = \frac{4}{3}\pi r^3 \]
Step 3: Detailed Explanation:
Let the first soap bubble have radius \( r_1 \), excess pressure \( P_1 \), and volume \( V_1 \).
Let the second soap bubble have radius \( r_2 \), excess pressure \( P_2 \), and volume \( V_2 \).
From the problem statement, the excess pressure inside the first bubble is 1.5 times that of the second:
\[ P_1 = 1.5 P_2 \]
Substitute the formula for excess pressure:
\[ \frac{4T}{r_1} = 1.5 \left( \frac{4T}{r_2} \right) \]
Assuming the bubbles are made of the same soap solution, the surface tension \( T \) is the same. We can cancel \( 4T \) from both sides:
\[ \frac{1}{r_1} = \frac{1.5}{r_2} \]
Rearranging to find the ratio of their radii:
\[ r_2 = 1.5 r_1 = \frac{3}{2} r_1 \]
Now, we are given that the volume of the second bubble is \( x \) times the volume of the first bubble:
\[ V_2 = x V_1 \]
Substitute the formula for the volume of a sphere:
\[ \frac{4}{3}\pi r_2^3 = x \left( \frac{4}{3}\pi r_1^3 \right) \]
Cancel the common geometric factors \( \frac{4}{3}\pi \):
\[ r_2^3 = x \cdot r_1^3 \]
Substitute the relationship between the radii \( r_2 = \frac{3}{2} r_1 \) into this equation:
\[ \left( \frac{3}{2} r_1 \right)^3 = x \cdot r_1^3 \]
\[ \frac{27}{8} r_1^3 = x \cdot r_1^3 \]
Cancel \( r_1^3 \) from both sides to solve for \( x \):
\[ x = \frac{27}{8} \]
Step 4: Final Answer:
The value of \( x \) is \( \frac{27}{8} \).