To find the ratio of the corresponding side lengths of two similar solids, given the ratio of their surface areas, we need to follow these steps:
1. Identify the ratio of the surface areas.
The given ratio of the surface areas of the two similar solids is [tex]\(16 : 144\)[/tex].
2. Express the ratio in fractional form.
The ratio [tex]\(16 : 144\)[/tex] can be written as:
[tex]\[
\frac{16}{144}
\][/tex]
3. Simplify the fraction.
To simplify [tex]\(\frac{16}{144}\)[/tex], we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 16 and 144 is 16:
[tex]\[
\frac{16 \div 16}{144 \div 16} = \frac{1}{9}
\][/tex]
4. Determine the side lengths ratio for similar solids.
For similar solids, the ratio of their side lengths corresponds to the square root of the ratio of their surface areas. Thus, we need to find the square root of [tex]\(\frac{1}{9}\)[/tex]:
[tex]\[
\sqrt{\frac{1}{9}} = \frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3}
\][/tex]
5. Convert the fraction to a ratio.
The fraction [tex]\(\frac{1}{3}\)[/tex] corresponds to the ratio [tex]\(1: 3\)[/tex]. However, the original surface area ratio was given as [tex]\(16: 144\)[/tex], so we must match this with our results.
6. Find the corresponding numbers in the original surface areas.
Since both the square root of the numerator and the denominator of the original surface area ratio are:
[tex]\[
\sqrt{16} = 4 \quad \text{and} \quad \sqrt{144} = 12
\][/tex]
These values confirm that the simplified side length ratio is:
[tex]\[
\frac{4}{12} \equiv 4: 12
\][/tex]
So, the ratio of their corresponding side lengths is:
[tex]\[
4: 12
\][/tex]
Hence, the correct option is:
A. [tex]\(4: 12\)[/tex]
