To find the scale factor of the side lengths of two similar squares given their areas, we need to follow these steps:
1. Understand the Relation Between Area and Side Length:
For a square, the area is equal to the side length squared. If the areas of two similar squares are given, the side lengths can be related through the areas.
2. Find the Areas of the Squares:
The areas of the two squares are given as [tex]\(16 \, \text{m}^2\)[/tex] and [tex]\(49 \, \text{m}^2\)[/tex].
3. Express the Scale Factor:
Similar squares have their side lengths in the same ratio as the square root of the ratio of their areas. If [tex]\(A_1\)[/tex] and [tex]\(A_2\)[/tex] are the areas of the two squares, and [tex]\(s_1\)[/tex] and [tex]\(s_2\)[/tex] are their respective side lengths, then the scale factor [tex]\(k\)[/tex] is given by:
[tex]\[
k = \sqrt{\frac{A_2}{A_1}}
\][/tex]
4. Calculate the Ratio of the Areas:
The ratio of the areas is:
[tex]\[
\frac{A_2}{A_1} = \frac{49}{16}
\][/tex]
5. Take the Square Root of the Ratio:
To find the scale factor [tex]\(k\)[/tex], take the square root of the ratio of the areas:
[tex]\[
k = \sqrt{\frac{49}{16}}
\][/tex]
6. Simplify the Square Root:
[tex]\[
\sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4} = 1.75
\][/tex]
Therefore, the scale factor of the side lengths of the two similar squares is [tex]\(1.75\)[/tex].
