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: Understand what is being asked.
Two spheres have volumes in the ratio $64:27$, and we need the ratio of their surface areas. Instead of directly comparing exponents of the ratio, let us assign actual radius variables to the spheres and compute the surface areas explicitly.

Step 2: Assign the radii using the volume ratio.
Since volume is proportional to the cube of the radius, and $64=4^3$ while $27=3^3$, we can write the radii as:
\[ r_1 = 4k, \quad r_2 = 3k \]
for some constant $k$, since then $\frac{r_1^3}{r_2^3}=\frac{(4k)^3}{(3k)^3}=\frac{64}{27}$ exactly matches the given ratio.

Step 3: Write the surface area of each sphere using these radii.
The surface area of a sphere of radius $r$ is $S=4\pi r^2$:
\[ S_1 = 4\pi(4k)^2 = 4\pi\times 16k^2 = 64\pi k^2 \]
\[ S_2 = 4\pi(3k)^2 = 4\pi\times 9k^2 = 36\pi k^2 \]
Step 4: Find the ratio.
\[ \frac{S_1}{S_2} = \frac{64\pi k^2}{36\pi k^2} = \frac{64}{36} = \frac{16}{9} \]
Final Answer:
The ratio of their surface areas is $16:9$, matching option (C).
\[ \boxed{16:9} \]