Question

The ratio of the volumes of two spheres is 64 : 125. Then the ratio of their surface areas is

• A. 25 : 8
• B. 25 : 16
• C. 16 : 25
• D. none of these

Hint:

The relationship is straightforward: to get from the volume ratio to the side/radius ratio, take the cube root. To get from the side/radius ratio to the area ratio, square it. So, cube root then square: \((\sqrt[3]{64})^2 : (\sqrt[3]{125})^2 = 4^2 : 5^2 = 16:25\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The volumes and surface areas of spheres are related to their radii. If we can find the ratio of the radii from the ratio of the volumes, we can then find the ratio of the surface areas.

Step 2: Key Formula or Approach:
Let the radii of the two spheres be \(r_1\) and \(r_2\).
Ratio of Volumes: \(\frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \left(\frac{r_1}{r_2}\right)^3\)
Ratio of Surface Areas: \(\frac{A_1}{A_2} = \frac{4\pi r_1^2}{4\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2\)

Step 3: Detailed Explanation:
We are given the ratio of volumes:
\[ \frac{V_1}{V_2} = \frac{64}{125} \] From this, we find the ratio of the radii:
\[ \left(\frac{r_1}{r_2}\right)^3 = \frac{64}{125} \] \[ \frac{r_1}{r_2} = \sqrt[3]{\frac{64}{125}} = \frac{4}{5} \] Now, we can find the ratio of the surface areas:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \] The ratio of their surface areas is 16 : 25.

Step 4: Final Answer:
The ratio of their surface areas is 16 : 25.