To find out which girl arrives at the activity center first, let's analyze each girl's distance function and calculate the time at which they arrive at the activity center, located 6 miles from the school.
Bethany's Distance Function
Bethany's distance is modeled by the equation: \[ y = 4x \] where:
- \( y \) is the distance in miles
- \( x \) is the time in hours
To find the time at which Bethany reaches the activity center (6 miles): \[ 6 = 4x \] \[ x = \frac{6}{4} = 1.5 \text{ hours} \]
Morgan's Distance Function
Morgan starts her journey 0.5 hours after Bethany, so her equation is: \[ y = 8x - 4 \] To find her arrival time taking into account her delayed start (and the fact she travels for \( x - 0.5 \)): If we let \( x \) be the total time from when Bethany starts:
- Morgan effectively starts at \( x = 0.5 \).
We need to find the time when Morgan reaches 6 miles: Set the equation for Morgan equal to 6: \[ 6 = 8(x - 0.5) \] \[ 6 = 8x - 4 \] \[ 10 = 8x \] \[ x = \frac{10}{8} = 1.25 \text{ hours} \] Since Morgan starts 0.5 hours later, her total travel time is: \[ 1.25 - 0.5 = 0.75 \text{ hours} \]
Taylor's Distance Function
Taylor starts her journey 1 hour after Bethany, and her distance function is given by: \[ y = 20x - 20 \] To find when Taylor reaches 6 miles (with her delayed start): Set the equation equal to 6: \[ 6 = 20(x - 1) \] \[ 6 = 20x - 20 \] \[ 26 = 20x \] \[ x = \frac{26}{20} = 1.3 \text{ hours} \] Since Taylor starts 1 hour later, her total travel time is: \[ 1.3 - 1 = 0.3 \text{ hours} \]
Summary of Arrival Times
- Bethany: Arrives in 1.5 hours.
- Morgan: Arrives in 1.25 hours (0.75 hours travel time).
- Taylor: Arrives in 1.3 hours (0.3 hours travel time).
Conclusion
Morgan arrives first, followed by Taylor, and Bethany arrives last. Therefore, the answer is:
Morgan.
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