To find the ratio of the corresponding side lengths of two similar solids given the ratio of their surface areas is [tex]\( 49: 100 \)[/tex], we can follow these steps:
1. Understanding the relationship between surface area and side length:
For similar solids, the ratio of their surface areas is equal to the square of the ratio of their corresponding side lengths. Let's denote the ratio of the side lengths as [tex]\( \frac{a}{b} \)[/tex].
2. Express the given surface area ratio:
The given ratio of the surface areas is [tex]\( 49: 100 \)[/tex]. This can be written as:
[tex]\[
\frac{\text{Surface Area of Solid 1}}{\text{Surface Area of Solid 2}} = \frac{49}{100}
\][/tex]
3. Using the relationship to find the side length ratio:
The ratio of side lengths is the square root of the ratio of surface areas. Hence:
[tex]\[
\left( \frac{a}{b} \right)^2 = \frac{49}{100}
\][/tex]
4. Take the square root of both sides:
[tex]\[
\frac{a}{b} = \sqrt{\frac{49}{100}} = \frac{\sqrt{49}}{\sqrt{100}} = \frac{7}{10}
\][/tex]
Therefore, the ratio of the corresponding side lengths is [tex]\( 7:10 \)[/tex].
So, the correct answer is:
A. [tex]\( 7:10 \)[/tex]
