To determine the location where Genevieve will make her cut on the ribbon, we need to follow these steps carefully, considering the parameters provided in the problem.
1. Understand the Ribbon Specifications:
- The total length of the ribbon is 60 inches.
- 2 inches on one end of the ribbon are frayed, so we will exclude these when making the cut.
2. Calculate the Effective Length of the Ribbon:
- The effective length is the total length minus the frayed portion:
[tex]\[
\text{Effective Length} = 60 \text{ inches} - 2 \text{ inches} = 58 \text{ inches}
\][/tex]
3. Given Ratio and Starting Point:
- The ribbon needs to be cut in a ratio of [tex]\(2:3\)[/tex].
- The fraying starts at 2 inches, meaning the starting point for our effective length is at 2 inches.
4. Setting up the Given Formula:
- The formula to find the cut location is:
[tex]\[
x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
Here:
- [tex]\(m = 2\)[/tex]
- [tex]\(n = 3\)[/tex]
- [tex]\(x_1\)[/tex] is the starting point, which is 2 inches.
- [tex]\(x_2\)[/tex] is the ending point, which is [tex]\(2 + \text{Effective Length} = 2 + 58 = 60\)[/tex] inches.
5. Plugging in the Values:
- Calculate the effective proportion for cutting based on the ratio:
[tex]\[
\frac{m}{m + n} = \frac{2}{2 + 3} = \frac{2}{5}
\][/tex]
- Now, calculate the cut location [tex]\(x\)[/tex]:
[tex]\[
x = \left(\frac{2}{5}\right) \left(60 - 2\right) + 2
\][/tex]
- Simplify inside the parentheses:
[tex]\[
x = \left(\frac{2}{5}\right) \times 58 + 2
\][/tex]
- Perform the multiplication:
[tex]\[
x = \left(\frac{2 \times 58}{5}\right) + 2 = \left(\frac{116}{5}\right) + 2 = 23.2 + 2
\][/tex]
- Add the starting point:
[tex]\[
x = 25.2
\][/tex]
Therefore, the cut will be located at 25.2 inches on the ribbon.
The correct answer is:
[tex]\[
\boxed{25.2 \text{ in.}}
\][/tex]
