To find the ratio of the corresponding side lengths of two similar solids when given the ratio of their surface areas, we need to follow these steps:
1. Understand the Relationship Between Surface Area and Side Lengths:
The ratio of the surface areas of two similar solids is the square of the ratio of their corresponding side lengths.
2. Given Ratio of Surface Areas:
The ratio of the surface areas is given as [tex]\( 16 : 144 \)[/tex].
3. Set Up and Simplify the Ratio:
Simplify the ratio of the surface areas:
[tex]\[
\frac{16}{144} = \frac{1}{9}
\][/tex]
4. Find the Ratio of Side Lengths:
Since the ratio of the surface areas is the square of the ratio of side lengths, we take the square root of [tex]\( \frac{1}{9} \)[/tex]:
[tex]\[
\text{Ratio of side lengths} = \sqrt{\frac{1}{9}} = \frac{1}{3}
\][/tex]
5. Express the Ratio in Whole Numbers:
To express this ratio as whole numbers, we multiply both terms by 12 (since [tex]\( \frac{1}{3} \times 12 = 4 \)[/tex]):
[tex]\[
\text{Ratio of side lengths} = 4 : 12
\][/tex]
So, the correct answer is:
[tex]\[ \boxed{4 : 12} \][/tex]
Thus, the ratio of their corresponding side lengths is [tex]\( 4:12 \)[/tex], which corresponds to option A.
