To find where Genevieve will make the cut on the 60-inch piece of ribbon, we'll go through the following steps:
1. Identify the total length of the ribbon and the frayed length.
- The total length of the ribbon is 60 inches.
- The frayed end is 2 inches.
2. Determine the usable length of the ribbon that is not frayed.
- The non-frayed length of the ribbon is [tex]\(60 - 2 = 58\)[/tex] inches.
3. Determine the starting point after the frayed end.
- The starting point is at 2 inches.
4. Identify the total length from the starting point to the end of the ribbon.
- This length is from [tex]\(2\)[/tex] inches to [tex]\(60\)[/tex] inches.
5. Use the given ratio [tex]\(2:3\)[/tex] to find where to cut the ribbon.
- To use the formula for the cut position in the ratio [tex]\( \frac{m}{m+n} \)[/tex], we need:
[tex]\[
m = 2 \quad \text{and} \quad n = 3
\][/tex]
6. Apply the length and ratio to the formula [tex]\( x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1 \)[/tex]:
- Here, [tex]\( x_1 = 2 \)[/tex] inches (starting point) and [tex]\( x_2 = 60 \)[/tex] inches (end of ribbon).
[tex]\[
x = \left(\frac{2}{2+3}\right) \left(60 - 2 \right) + 2
\][/tex]
[tex]\[
x = \left(\frac{2}{5}\right) \cdot 58 + 2
\][/tex]
[tex]\[
x = 0.4 \cdot 58 + 2
\][/tex]
[tex]\[
x = 23.2 + 2
\][/tex]
[tex]\[
x = 25.2 \text{ inches}
\][/tex]
7. Round the result to the nearest tenth:
- From the calculation, the cut position is [tex]\( 25.2 \text{ inches} \)[/tex].
So, Genevieve should make the cut at 25.2 inches from the starting point of the ribbon. Among the given options, 25.2 inches is the correct location.
