To find the equation that shows the proportional relationship between the distance \( d \) and the time \( t \) for the train, we can start by calculating the speed of the train.
The distance traveled is 67.5 miles in 1.5 hours. To find the speed (which is the distance per unit of time), we can use the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{67.5 \text{ miles}}{1.5 \text{ hours}} = 45 \text{ miles per hour} \]
Since the distance \( d \) traveled by the train is directly proportional to the time \( t \), we can express this relationship in the form of the equation:
\[ d = \text{Speed} \times t \]
Substituting the speed we calculated, we have:
\[ d = 45t \]
So, among the options provided, the correct equation that shows the proportional relationship between the distance \( d \) and the time \( t \) that the train has traveled is:
\[ d = 45t \]
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