Question

Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is :

• A. 22 : 7
• B. 11 : 14
• C. 16 : 9
• D. 9 : 16

Hint:

For any similar 3D shapes, the ratio of areas is always the cube root of the volume ratio squared:
\[ \text{Area Ratio} = (\sqrt[3]{\text{Volume Ratio}})^2 \]
With a volume ratio of \(64:27\), take the cube root to get \(4:3\), then square it to get \(16:9\) instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is from Mensuration, focusing on the scaling properties of spheres.
We are given the ratio of the volumes of two spheres, and we need to determine the ratio of their surface areas.

Step 2: Key Formula or Approach:
Let the radii of the two spheres be \(R_1\) and \(R_2\).
1. The volume \(V\) of a sphere scales with the cube of its radius:
\[ \frac{V_1}{V_2} = \left(\frac{R_1}{R_2}\right)^3 \]
2. The surface area \(A\) of a sphere scales with the square of its radius:
\[ \frac{A_1}{A_2} = \left(\frac{R_1}{R_2}\right)^2 \]

Step 3: Detailed Explanation:
We are given the volume ratio:
\[ \frac{V_1}{V_2} = \frac{64}{27} \]
Using the volume-radius scaling relationship:
\[ \left(\frac{R_1}{R_2}\right)^3 = \frac{64}{27} \]
Take the cube root of both sides to find the ratio of the radii:
\[ \frac{R_1}{R_2} = \sqrt[3]{\frac{64}{27}} \]
Since \(64 = 4^3\) and \(27 = 3^3\):
\[ \frac{R_1}{R_2} = \frac{4}{3} \]
Now, compute the ratio of their surface areas:
\[ \frac{A_1}{A_2} = \left(\frac{R_1}{R_2}\right)^2 \]
Substitute the radius ratio:
\[ \frac{A_1}{A_2} = \left(\frac{4}{3}\right)^2 \]
\[ \frac{A_1}{A_2} = \frac{16}{9} \]

Step 4: Final Answer:
The ratio of their surface areas is 16 : 9.