To determine which girl arrives at the activity center first, we need to analyze the distance equations for each girl and see when they reach the six-mile mark.
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Bethany: \[ y = 4x \] Here, \( y \) represents the distance in miles, and \( x \) represents the time in hours. To find when she arrives at the center, we set \( y = 6 \): \[ 6 = 4x \implies x = \frac{6}{4} = 1.5 \text{ hours} \] Since she leaves right after school, she arrives at \( 1.5 \) hours after school starts.
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Morgan: \[ y = 8x - 4 \] Morgan leaves half an hour after Bethany, so she effectively starts at \( x = 0.5 \). To find when she arrives, we set \( y = 6 \): \[ 6 = 8x - 4 \implies 8x = 10 \implies x = \frac{10}{8} = 1.25 \text{ hours} \] However, since she starts at \( x = 0.5 \), her actual time after school is: \[ 1.25 + 0.5 = 1.75 \text{ hours} \]
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Taylor: \[ y = 20x - 20 \] Taylor leaves one hour after Bethany, which means she starts at \( x = 1 \). Setting \( y = 6 \): \[ 6 = 20x - 20 \implies 20x = 26 \implies x = \frac{26}{20} = 1.3 \text{ hours} \] Since she started one hour later, her actual time after school is: \[ 1.3 + 1 = 2.3 \text{ hours} \]
Now, comparing the arrival times after school:
- Bethany: \( 1.5 \) hours
- Morgan: \( 1.75 \) hours
- Taylor: \( 2.3 \) hours
Conclusion: Bethany arrives first at the activity center.
The correct response is Bethany.
1 answer