To find the ratio of the corresponding side lengths of the two similar solids, given that the ratio of their surface areas is \(49:100\), we can follow these steps:
1. Understand the relationship between surface area and side length in similar solids:
- For similar solids, the ratio of their surface areas is the square of the ratio of their corresponding side lengths.
2. Express the given information:
- The ratio of the surface areas is given as \(49:100\).
- Let the ratio of the side lengths be \(a:b\).
3. Set up the relationship using the given ratio:
- Since the ratio of the surface areas is \(49:100\), this can be written as:
[tex]\[
\left(\frac{a}{b}\right)^2 = \frac{49}{100}
\][/tex]
4. Solve for the ratio of the side lengths:
- To find the ratio of the side lengths, we need to take the square root of both sides:
[tex]\[
\frac{a}{b} = \sqrt{\frac{49}{100}}
\][/tex]
- The square root of 49 is 7.
- The square root of 100 is 10.
- Therefore, the ratio of the side lengths is:
[tex]\[
\frac{a}{b} = \frac{7}{10}
\][/tex]
5. Express the ratio in simplest form:
- The simplest form of the ratio \(7:10\) is just \(7:10\).
Therefore, the correct answer is:
[tex]\[
\boxed{A. \; 7:10}
\][/tex]
