To find out which girl arrives at the activity center first, we need to determine when the distance traveled by each girl reaches 6 miles.
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Bethany:
- Her equation is \( y = 4x \).
- Set \( y = 6 \) to find out when she arrives: \[ 6 = 4x \ x = \frac{6}{4} = 1.5 \text{ hours} \]
- Bethany arrives at the activity center 1.5 hours after she leaves.
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Morgan:
- Morgan leaves half an hour after Bethany, meaning she effectively starts after \( x = 0.5 \).
- Her equation is \( y = 8x - 4 \).
- However, we need to adjust her \( x \) value to account for her late start. Her effective time is \( x - 0.5 \).
- So, we set up her equation: \[ y = 8(x - 0.5) - 4 \ y = 8x - 4 - 4 \ y = 8x - 8 \]
- Now, set \( y = 6 \): \[ 6 = 8x - 8 \ 14 = 8x \ x = \frac{14}{8} = 1.75 \text{ hours} \]
- Morgan arrives at the activity center 1.75 hours after Bethany leaves.
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Taylor:
- Taylor leaves one hour after Bethany, meaning she effectively starts after \( x = 1 \).
- Her equation is \( y = 20x - 20 \).
- For Taylor, we adjust \( x \) to account for her late start. Her actual travel time is \( x - 1 \).
- Set up her equation: \[ y = 20(x - 1) - 20 \ y = 20x - 20 - 20 \ y = 20x - 40 \]
- Now, set \( y = 6 \): \[ 6 = 20x - 40 \ 46 = 20x \ x = \frac{46}{20} = 2.3 \text{ hours} \]
- Taylor arrives at the activity center 2.3 hours after Bethany leaves.
Now we can compare:
- Bethany: 1.5 hours
- Morgan: 1.75 hours
- Taylor: 2.3 hours
Therefore, Bethany arrives at the activity center first. The correct response is Bethany.