Question

What is the ratio of the surface area of two soap bubbles if their radii are in the ratio \(2:3\)?

• A. \(2:3\)
• B. \(4:9\)
• C. \(3:2\)
• D. \(9:4\)

Hint:

Surface area of a sphere varies as the square of the radius. So if \(r_1:r_2 = a:b\), then \(A_1:A_2 = a^2:b^2\).

Answer 1

The Correct Option is B

Concept: The surface area of a sphere (or soap bubble) is given by \[ A = 4\pi r^2 \] Thus, surface area is proportional to the square of the radius. \[ A \propto r^2 \]

Step 1: Write the ratio of radii. \[ r_1 : r_2 = 2 : 3 \]

Step 2: Find the ratio of surface areas. Since \(A \propto r^2\), \[ A_1 : A_2 = r_1^2 : r_2^2 \] \[ = 2^2 : 3^2 \] \[ = 4 : 9 \] Thus, \[ \boxed{4:9} \]