To find out when Sara’s train will meet up with Debbie's train, we need to determine the distance each train has traveled over time and find the point at which those distances are equal.
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Debbie's Train Equation:
- The equation for Debbie's train is given as \( y = 40x + 10 \), where \( y \) is the distance traveled in miles, and \( x \) is the time in hours after Debbie left.
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Sara's Train Equation:
- The equation for Sara’s train is \( y = 60x - 10 \), but since Sara leaves 10 minutes after Debbie, we must adjust the time for Sara's departure.
- Since 10 minutes is \( \frac{1}{6} \) of an hour, we will set \( x \) to be the time in hours since Debbie left, but Sara will start traveling after \( \frac{1}{6} \) hours. Therefore, if \( t \) is the time in hours after Debbie leaves, the equation for Sara's travel becomes \( y = 60(t - \frac{1}{6}) - 10 \).
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Simplifying Sara's Train Equation: \[ y = 60(t - \frac{1}{6}) - 10 = 60t - 10 - 10 = 60t - 20 \]
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Setting the Two Equations Equal: Now we need to find the time \( t \) when both distances are equal: \[ 40t + 10 = 60t - 20 \]
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Solving the Equation: \[ 10 + 20 = 60t - 40t \] \[ 30 = 20t \] \[ t = \frac{30}{20} = 1.5 \text{ hours} \]
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Converting Time to Minutes: Since \( t \) was in hours, we convert it to minutes: \[ 1.5 \text{ hours} = 1.5 \times 60 = 90 \text{ minutes} \]
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Conclusion: Therefore, Sara's train will meet up with Debbie's train 90 minutes after Debbie's departure.
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