To determine which girl arrives at the activity center first, we need to find the time it takes each girl to reach the activity center based on their distance equations.
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Bethany's Distance Equation:
- Equation: \( y = 4x \)
- Here, \( y \) represents the distance traveled in miles, and \( x \) represents the time in hours since she left school.
- She travels at a speed of 4 miles per hour.
To find the time she takes to travel 6 miles: \[ 6 = 4x \implies x = \frac{6}{4} = 1.5 \text{ hours} \] So, Bethany arrives at the activity center 1.5 hours after leaving school.
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Morgan's Distance Equation:
- Equation: \( y = 8x - 4 \)
- Morgan leaves half an hour after Bethany, meaning if Bethany starts at \( t = 0 \), Morgan starts at \( t = 0.5 \) hours.
- She travels at a speed of 8 miles per hour.
To find the time she takes to travel 6 miles: \[ 6 = 8x - 4 \implies 10 = 8x \implies x = \frac{10}{8} = 1.25 \text{ hours (since she starts at 0.5 hours, her total time to reach is 1.25 + 0.5 = 1.75 hours)} \] Morgan arrives at the activity center 1.75 hours after Bethany leaves (1.25 hours traveling + 0.5 hours waiting).
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Taylor's Distance Equation:
- Equation: \( y = 20x - 20 \)
- Taylor leaves one hour after Bethany, so if Bethany starts at \( t = 0 \), Taylor starts at \( t = 1 \) hour.
- She travels at a speed of 20 miles per hour.
To find her time to travel 6 miles: \[ 6 = 20x - 20 \implies 26 = 20x \implies x = \frac{26}{20} = 1.3 \text{ hours (since she starts at hour 1, her total time to reach is 1.3 + 1 = 2.3 hours)} \] Taylor arrives at the activity center 2.3 hours after Bethany leaves (1.3 hours traveling + 1 hour waiting).
Summary of Arrival Times:
- Bethany: 1.5 hours after leaving
- Morgan: 1.75 hours after leaving
- Taylor: 2.3 hours after leaving
Therefore, Bethany arrives at the activity center first, in 1.5 hours after she leaves school.