To determine the ratio of the corresponding side lengths of two similar solids when the ratio of their surface areas is given, we need to follow these steps:
1. Understanding the Given Ratio:
The ratio of the surface areas of the two similar solids is given as [tex]\(49:100\)[/tex]. This can be written as a fraction:
[tex]\[
\text{Surface area ratio} = \frac{49}{100}
\][/tex]
2. Relating 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]. Therefore, we have:
[tex]\[
\left(\frac{a}{b}\right)^2 = \frac{49}{100}
\][/tex]
3. Finding the Ratio of Side Lengths:
To find the actual ratio of side lengths ([tex]\(\frac{a}{b}\)[/tex]), we need to take the square root of both sides of the surface area ratio equation:
[tex]\[
\frac{a}{b} = \sqrt{\frac{49}{100}}
\][/tex]
Calculate the square root of the numerator and the denominator separately:
[tex]\[
\sqrt{49} = 7 \quad \text{and} \quad \sqrt{100} = 10
\][/tex]
So, the ratio of the side lengths is:
[tex]\[
\frac{a}{b} = \frac{7}{10}
\][/tex]
4. 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].
Thus, the correct answer is:
C. [tex]\(7: 10\)[/tex]
