To determine the ratio of the side lengths of two similar solids given the ratio of their surface areas, we follow these steps:
1. Identify the given ratio of the surface areas.
The ratio of the surface areas is given as [tex]\( 16 : 144 \)[/tex].
2. Simplify the ratio of the surface areas.
We can simplify the ratio [tex]\( 16 : 144 \)[/tex] by dividing both numbers by their greatest common divisor, which is 16:
[tex]\[
\frac{16}{144} = \frac{16 \div 16}{144 \div 16} = \frac{1}{9}
\][/tex]
Hence, the simplified ratio of the surface areas is [tex]\( 1 : 9 \)[/tex].
3. Understand the relation between surface area and side length ratios for similar solids.
For similar solids, the ratio of their surface areas is the square of the ratio of their corresponding side lengths. Let the side length ratio be [tex]\( \left(\frac{a}{b}\right)^2 = \frac{1}{9} \)[/tex].
4. Find the side length ratio.
To find the side length ratio, we take the square root of both sides:
[tex]\[
\sqrt{\left(\frac{a}{b}\right)^2} = \sqrt{\frac{1}{9}}
\][/tex]
Simplifying this,
[tex]\[
\frac{a}{b} = \frac{1}{3}
\][/tex]
Therefore, the ratio of the corresponding side lengths is [tex]\( 1 : 3 \)[/tex].
5. Express the ratio in simplest form.
We can multiply both sides of [tex]\( 1 : 3 \)[/tex] by 4 to maintain the same ratio properties observed between simplified areas and side lengths:
[tex]\[
1 : 3 \quad \text{can be scaled} \quad 4 : 12
\][/tex]
Therefore, the most suitable answer from the given options for the ratio of the side lengths of the two similar solids is:
B. [tex]\( 4 : 12 \)[/tex]
