If you have two similar objects that have a scale factor of 6:11, what is the ratio of their surface areas?

Break Down the Problem

1. Identify the scale factor for the two similar objects: 6:11.
2. Understand that the scale factor applies to the linear dimensions for similar objects, and that the ratio of the surface areas will be based on the square of the scale factor.

Relevant Concepts

3. The formula for the ratio of the surface areas $R$ of two similar shapes is given by: $R = \left( \frac{\text{Scale Factor of Object 1}}{\text{Scale Factor of Object 2}} \right)^2$ In this case, the scale factor is given as $\frac{6}{11}$.

Analysis and Detail

4. Calculate the ratio of their surface areas: $R = \left( \frac{6}{11} \right)^2$ $R = \frac{36}{121}$

Verify and Summarize

5. Verify that the calculation is correct and confirms that the ratio of the surface areas corresponds to the square of the linear ratio.
6. Summarize that the surface area ratio of the two similar objects is $\frac{36}{121}$.

Final Answer

The ratio of their surface areas is $\frac{36}{121}$.