To determine where Genevieve will make the cut on a 60-inch piece of ribbon, we need to find the cut location considering the ratio [tex]\(2:3\)[/tex] and the 2 inches of the ribbon that are frayed at one end.
1. Determine the effective length:
The total length of the ribbon is 60 inches, but the first 2 inches are frayed and should not be considered when making the cut. Thus, the effective length of the ribbon to be used is:
[tex]\[
\text{Effective Length} = 60 \text{ inches} - 2 \text{ inches} = 58 \text{ inches}
\][/tex]
2. Understand the ratio [tex]\(2:3\)[/tex]:
The ratio [tex]\(2:3\)[/tex] means that when this effective length is split into two parts, one part will be [tex]\(\frac{2}{5}\)[/tex] of the effective length, and the other part will be [tex]\(\frac{3}{5}\)[/tex] of the effective length, where the total parts [tex]\(2 + 3 = 5\)[/tex].
3. Locate the cut using the given formula:
The formula given:
[tex]\[
x = \left( \frac{m}{m + n} \right) (x_2 - x_1) + x_1
\][/tex]
where:
- [tex]\(m = 2\)[/tex]
- [tex]\(n = 3\)[/tex]
- [tex]\(x_1 = 2\)[/tex] (the starting point after the frayed end)
- [tex]\(x_2 = 60\)[/tex] (the total length of the ribbon including the frayed part)
Substituting these values into the formula:
[tex]\[
x = \left( \frac{2}{2 + 3} \right) (60 - 2) + 2
\][/tex]
Calculate each step:
[tex]\[
x = \left( \frac{2}{5} \right) \cdot 58 + 2
\][/tex]
[tex]\[
x = \left( \frac{2 \cdot 58}{5} \right) + 2
\][/tex]
[tex]\[
x = \left( \frac{116}{5} \right) + 2
\][/tex]
[tex]\[
x = 23.2 + 2 = 25.2
\][/tex]
4. Round to the nearest tenth:
The computed location is already at [tex]\(25.2\)[/tex], which is already rounded to the nearest tenth.
Therefore, the cut location where Genevieve should make the cut is:
[tex]\[
\boxed{25.2}
\][/tex]
Among the given options, the correct answer is:
[tex]\[
25.2 \text{ in.}
\][/tex]
