Given that the ratio of the surface areas of two similar solids is [tex]\( 49 : 100 \)[/tex], we need to find the ratio of their corresponding side lengths.
To determine this, let's follow these steps:
1. Understanding the relationship between surface area and side lengths:
- For similar solids, the ratio of their surface areas is 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].
- If [tex]\(\text{Surface Area Ratio} = \left(\frac{a}{b}\right)^2\)[/tex], then [tex]\( \frac{a^2}{b^2} = \frac{49}{100} \)[/tex].
2. Taking the square root of the surface area ratio:
- To find the ratio of the side lengths, we take the square root of the surface area ratio [tex]\( \frac{49}{100} \)[/tex].
- [tex]\(\sqrt{\frac{49}{100}} = \frac{\sqrt{49}}{\sqrt{100}} = \frac{7}{10}\)[/tex].
3. Simplifying the ratio:
- The ratio [tex]\(\frac{7}{10}\)[/tex] is already in its simplest form.
Therefore, the ratio of the corresponding side lengths of the two similar solids is [tex]\( 7 : 10 \)[/tex].
So, the correct answer is [tex]\( \boxed{7 : 10} \)[/tex].
