Certainly! To solve this problem, we need to use the fundamental relationship between distance, speed, and time, which is given by the equation:
[tex]\[ \text{Distance} = \text{Speed} \times \text{Time} \][/tex]
First, let's identify the information given:
1. The speed of the eastbound train is 75 miles per hour.
2. The speed of the westbound train is 55 miles per hour.
3. We want to find out how long it will take for the two trains to be 312 miles apart.
Since the two trains are traveling in opposite directions, their speeds will add up. Let [tex]\( t \)[/tex] represent the time in hours that it takes for the two trains to be 312 miles apart.
The combined speed of both trains will be:
[tex]\[ 75 + 55 = 130 \text{ miles per hour} \][/tex]
Now, we can use the distance formula to find the time:
[tex]\[ \text{Distance} = \text{Combined Speed} \times \text{Time} \][/tex]
So, we set up the equation with the given distance:
[tex]\[ 312 = 130 \times t \][/tex]
Next, solve for [tex]\( t \)[/tex] by dividing both sides of the equation by the combined speed:
[tex]\[ t = \frac{312}{130} \][/tex]
Now, perform the division to find [tex]\( t \)[/tex]:
[tex]\[ t = 2.4 \text{ hours} \][/tex]
Therefore, it will take 2.4 hours for the two trains to be 312 miles apart.
