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

Step 1: Understanding the Question:
We are given the ratio of the radii of two spherical soap bubbles and asked to find the ratio of their surface areas.
Step 2: Key Formula or Approach:
The surface area (\(A\)) of a sphere with radius \(r\) is given by the formula \(A = 4\pi r^2\).
This formula shows that the surface area is directly proportional to the square of the radius, i.e., \(A \propto r^2\).
Step 3: Detailed Explanation:
Let the radii of the two soap bubbles be \(r_1\) and \(r_2\). We are given their ratio:
\[ \frac{r_1}{r_2} = \frac{2}{3} \] Let their respective surface areas be \(A_1\) and \(A_2\). The formula for surface area is \(A_1 = 4\pi r_1^2\) and \(A_2 = 4\pi r_2^2\).
To find the ratio of their surface areas, we divide \(A_1\) by \(A_2\):
\[ \frac{A_1}{A_2} = \frac{4\pi r_1^2}{4\pi r_2^2} = \frac{r_1^2}{r_2^2} \] This can be written as:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 \] Now, substitute the given ratio of the radii:
\[ \frac{A_1}{A_2} = \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \] Step 4: Final Answer:
The ratio of the surface areas is \(4:9\).