To determine which girl arrives first at the beach, we can analyze each girl's distance function and their respective departure times.
Equations Analysis:
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B's Equation: \( y = 4x \)
- This means that B travels at a speed of 4 miles per hour.
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M's Equation: \( y = 8x - 4 \)
- This indicates that M travels at a speed of 8 miles per hour.
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T's Equation: \( y = 20x - 20 \)
- T travels at a speed of 20 miles per hour.
Departure Times:
- B leaves at \( t = 0 \) hours.
- M leaves at \( t = 0.5 \) hours.
- T leaves at \( t = 1 \) hour.
Arrival Times Calculation:
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B's Arrival Time:
- Distance to beach is 6 miles.
- Using \( y = 4x \) for B, we set \( y = 6 \): \[ 6 = 4x \Rightarrow x = \frac{6}{4} = 1.5 \text{ hours} \]
- Therefore, B arrives at \( 0 + 1.5 = 1.5 \) hours.
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M's Arrival Time:
- M leaves at \( t = 0.5 \).
- For M, using \( y = 8x - 4 \): \[ 6 = 8x - 4 \Rightarrow 8x = 10 \Rightarrow x = \frac{10}{8} = 1.25 \text{ hours} \]
- Total time from the start of B's journey: \[ 0.5 + 1.25 = 1.75 \text{ hours} \]
- Therefore, M arrives at \( 1.75 \) hours.
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T's Arrival Time:
- T leaves at \( t = 1 \).
- For T, using \( y = 20x - 20 \): \[ 6 = 20x - 20 \Rightarrow 20x = 26 \Rightarrow x = \frac{26}{20} = 1.3 \text{ hours} \]
- Total time from the start of B's journey: \[ 1 + 1.3 = 2.3 \text{ hours} \]
- Therefore, T arrives at \( 2.3 \) hours.
Conclusion:
- B's arrival time: 1.5 hours
- M's arrival time: 1.75 hours
- T's arrival time: 2.3 hours
Thus, B arrives first at 1.5 hours.