To determine how far the car will have to travel to catch up to the train, we can start by calculating how far the train has traveled by the time the car starts.
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Calculate the train's distance when the car starts: The train leaves 2 hours earlier than the car and travels at a speed of 40 miles per hour. Therefore, in 2 hours, the train travels: \[ \text{Distance}_{\text{train}} = \text{Speed} \times \text{Time} = 40 , \text{miles/hour} \times 2 , \text{hours} = 80 , \text{miles} \]
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Let \( t \) be the time in hours that the car travels until it catches up to the train. During this time, the train will continue to travel. The distance traveled by the train during the time \( t \) is: \[ \text{Distance}_{\text{train}} = 40 , \text{miles/hour} \times (t + 2) , \text{hours} \]
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Distance traveled by the car: The distance that the car travels in \( t \) hours at 60 miles per hour is: \[ \text{Distance}_{\text{car}} = 60 , \text{miles/hour} \times t , \text{hours} \]
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Setting the distances equal: At the point when the car catches up to the train, both distances will be equal: \[ \text{Distance}{\text{train}} = \text{Distance}{\text{car}} \] \[ 40 , (t + 2) = 60t \]
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Solving for \( t \): Expanding the left side gives: \[ 40t + 80 = 60t \] Rearranging the equation leads to: \[ 80 = 60t - 40t \] \[ 80 = 20t \] \[ t = \frac{80}{20} = 4 , \text{hours} \]
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Calculating the distance the car travels: Now, we will find the distance the car has traveled in that time: \[ \text{Distance}_{\text{car}} = 60 , \text{miles/hour} \times 4 , \text{hours} = 240 , \text{miles} \]
Therefore, the car will have to travel 240 miles to catch up to the train.
The answer is E. 240 miles.