Where will Genevieve's cut be located? Round to the nearest tenth.

Given:
[tex]\[ x = \left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1 \][/tex]

Genevieve is cutting a 60-inch piece of ribbon into a ratio of [tex]\(2:3\)[/tex]. Since 2 inches are frayed at one end of the ribbon, she will need to start 2 inches in. This is indicated as 2 on the number line.

Options:
A. 25.2 in.
B. 29.4 in.
C. 35.1 in.
D. 40.7 in.



Answer :

To determine where Genevieve should cut the ribbon, we'll follow the given formula and the described situation step by step.

### Step-by-Step Solution

1. Understand the Problem:
Genevieve needs to cut a 60-inch piece of ribbon into segments with a ratio of [tex]\( 2:3 \)[/tex]. There is 2 inches of frayed ribbon which she won't use, starting to measure from the 2-inch mark.

2. Given Values:
- Total length of the ribbon: 60 inches
- Ratio values: [tex]\( m = 2 \)[/tex], [tex]\( n = 3 \)[/tex]
- Starting point: [tex]\( x_1 = 2 \)[/tex] inches from one end (the part of the ribbon that is frayed)

3. Calculate the Total Ratio Parts:
The total ratio parts [tex]\( m + n = 2 + 3 = 5 \)[/tex].

4. Calculate the Effective Ribbon Length:
Since the first 2 inches are frayed and can't be used, we work with the remaining:
[tex]\[ x_2 - x_1 = 60 - 2 = 58 \text{ inches} \][/tex]

5. Calculate the Fraction of the Ribbon for Segment [tex]\( m \)[/tex]:
The fraction of the ribbon for segment [tex]\( m \)[/tex] is:
[tex]\[ \frac{m}{m+n} = \frac{2}{5} = 0.4 \][/tex]

6. Find the Location of the Cut:
Use the formula [tex]\( x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1 \)[/tex]:
\begin{align}
x &= \left(0.4 \right) \left( 58 \right) + 2 \\
&= 23.2 + 2 \\
&= 25.2
\end{align
}

So, the location of the cut is at [tex]\( 25.2 \)[/tex] inches.

### Final Answer:
Genevieve should make the cut at 25.2 inches, rounded to the nearest tenth.

Therefore, the correct option is:
[tex]\[ \boxed{25.2 \text{ in}} \][/tex]

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