Help :C Two cities are 45 miles apart. Two trains, traveling at speeds of 70 mph and 60 mph, leave the two cities at the same time and travel in the same direction so that the faster train will catch up to the other one. How long after the trains leave will the distance between them be 10 miles? How long after the trans leave will they be 10 miles apart for the first time? Help :(

Explanation:

Draw a horizontal number line. Plot point A at 0 and point B at 45 on the number line. The distance from A to B is 45 units to represent that 45 mile gap between the cities.

Let's say the trains travel to the right (aka to the east) along this number line train track. We'll place the faster train at location A, and the slower train at location B. This allows the slower train a head start. Eventually the faster train will catch up. If this was swapped and the trains still aimed to the right, then the trains will never be 10 miles apart.

Now we'll define two variables

For the faster train at location A, which I'll just refer to "train A" from now on, we can form the equation y = 70x. I'm using the idea that distance = rate*time to help form this equation.

For example, if x = 2 hours have passed by, then train A is at location y = 70*2 = 140 miles on the number line train track.

Train B will have the equation y = 60x+45. The 60x is formed in a similar manner compared to the 70x. The plus 45 at the end is due to the 45 mile head start. In other words, train B travels 60x miles on top of the 45 miles already done so far.

The two key takeaways from that last section is that we have these equations

which correspond to the locations of train A and train B in that order.

Now to the question at hand: how can we determine when the trains are 10 miles apart? The short answer is through subtraction. Moreover, we'll need to involve absolute value to ensure that distance is never negative.

So the distance expression between the trains at any given time x is:

distance between trains = |trainA - trainB|

distance = |70x - (60x+45)|

distance = |70x-60x-45|

distance = |10x-45|

Next, we set that distance expression equal to 10 and solve for x.

distance = 10 miles

|10x-45| = 10

10x-45 = 10 or 10x-45 = -10

10x = 10+45 or 10x = -10+45

10x = 55 or 10x = 35

x = 55/10 or x = 35/10

x = 5.5 or x = 3.5

Those two results mean that at the hour marks of 3.5 hours and 5.5 hours is when the two trains are 10 miles apart exactly.

The two trains start off being 45 miles apart. They both take off and train A will catch up to train B. Their gap will be 10 miles at the 3.5 hour mark. That gap shrinks to 0 at some point, and then grows again afterward. By the 5.5 hour mark is when the trains are again 10 miles apart. This time is when train A is in the lead and will remain so assuming the speeds stay the same.  

It seems like your teacher is only concerned with when the trains are 10 miles apart for the first time and not the second. So your teacher is likely only expecting one answer and that would be 3.5 hours

Side notes: