To determine which girl arrives at the activity center first, we can analyze the equations given for each girl's distance as a function of time.
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Bethany's Distance:
\( y = 4x \)- This means Bethany walks at a speed of 4 miles per hour. She starts at \( x = 0 \) (time after school).
To find the time it takes for Bethany to reach the activity center (6 miles), we set \( y = 6 \) in her equation: \[ 6 = 4x \implies x = \frac{6}{4} = 1.5 \text{ hours} \]
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Morgan's Distance:
\( y = 8x - 4 \)- Morgan rides her bike at a speed of 8 miles per hour but starts half an hour later than Bethany.
To find the time it takes for Morgan to reach the center, we set \( y = 6 \): \[ 6 = 8x - 4 \implies 10 = 8x \implies x = \frac{10}{8} = 1.25 \text{ hours} \] Since she starts at \( x = 0.5\), the total time from the school would be: \[ 1.25 + 0.5 = 1.75 \text{ hours} \]
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Taylor's Distance:
\( y = 20x - 20 \)- Taylor takes the bus and leaves one hour after Bethany.
To find the time it takes for Taylor to reach the center: \[ 6 = 20x - 20 \implies 26 = 20x \implies x = \frac{26}{20} = 1.3 \text{ hours} \] Since she starts at \( x = 1 \), the total time from the school would be: \[ 1.3 + 1 = 2.3 \text{ hours} \]
Now comparing the total times from the start:
- Bethany: 1.5 hours (arrives at 1.5 hours)
- Morgan: 1.75 hours (arrives at 1.75 hours)
- Taylor: 2.3 hours (arrives at 2.3 hours)
From this analysis, Bethany arrives at the activity center first.
Answer: Bethany