To determine which girl arrives at the activity center first, we can analyze the equations provided for each girl's travel distance over time.
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Bethany's distance equation: \[ y = 4x \] This means Bethany travels 4 miles per hour. To find out when she reaches the activity center (6 miles away): \[ 6 = 4x \implies x = \frac{6}{4} = 1.5 \text{ hours} \] Bethany arrives after 1.5 hours.
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Morgan's distance equation: \[ y = 8x - 4 \] Morgan travels at 8 miles per hour, but she leaves half an hour after Bethany (0.5 hours later). Thus, we need to adjust \(x\) by 0.5 hours: \[ 6 = 8(x - 0.5) \implies 6 = 8x - 4 \implies 8x = 10 \implies x = \frac{10}{8} = 1.25 \text{ hours} \] Since she leaves half an hour after Bethany, her total travel time from the time school ends is \(1.25 + 0.5 = 1.75\) hours.
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Taylor's distance equation: \[ y = 20x - 20 \] Taylor travels at 20 miles per hour, but she leaves one hour after Bethany. Therefore, we adjust \(x\) for her as follows: \[ 6 = 20(x - 1) \implies 6 = 20x - 20 \implies 20x = 26 \implies x = \frac{26}{20} = 1.3 \text{ hours} \] Since she leaves one hour after Bethany, her total travel time from the time school ends is \(1.3 + 1 = 2.3\) hours.
Now we summarize the arrival times based on the calculations:
- Bethany arrives in 1.5 hours.
- Morgan arrives in 1.75 hours.
- Taylor arrives in 2.3 hours.
Conclusion: Bethany arrives at the activity center first. Therefore, the answer is:
Bethany
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