Question

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

• A. 4 : 3
• B. 3 : 4
• C. 16 : 9
• D. 9 : 16

Hint:

For any similar 3D figures (including spheres, cubes, and similar cylinders):
If the scale factor of 1D dimensions (radii, heights) is \(k\):
- Ratio of areas is \(k^2\)
- Ratio of volumes is \(k^3\)
Since the volume ratio is given as \(64:27\), the 1D ratio is \(\sqrt[3]{64}:\sqrt[3]{27} = 4:3\).
Squaring this gives the area ratio directly: \(4^2 : 3^2 = 16:9\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given that the volumes of two spheres are in the ratio of \(64 : 27\). We need to determine the ratio of their surface areas.

Step 2: Key Formula or Approach:
1. The volume \(V\) of a sphere of radius \(r\) is given by:
\[ V = \frac{4}{3}\pi r^3 \]
Therefore, the ratio of the volumes of two spheres with radii \(r_1\) and \(r_2\) is:
\[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 \]
2. The surface area \(S\) of a sphere is given by:
\[ S = 4\pi r^2 \]
The ratio of their surface areas is:
\[ \frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2 \]

Step 3: Detailed Explanation:
1. Let the radii of the two spheres be \(r_1\) and \(r_2\) respectively.
2. Set up the volume ratio:
\[ \frac{V_1}{V_2} = \frac{64}{27} \]
\[ \left(\frac{r_1}{r_2}\right)^3 = \frac{64}{27} \]
3. Take the cube root on both sides to find the ratio of their radii:
\[ \frac{r_1}{r_2} = \sqrt[3]{\frac{64}{27}} = \frac{4}{3} \]
4. Now, find the ratio of their surface areas:
\[ \frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2 \]
\[ \frac{S_1}{S_2} = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
Thus, the surface areas are in the ratio \(16 : 9\).

Step 4: Final Answer:
The ratio of their surface areas is \(16 : 9\), which corresponds to option (C).