Application problem

Question

A train travels east from Chicago at 6pm at 45 mph (EST. A train travels west from Cleveland at 35 mph at 7 pm (EST). If the cities are 285 miles apart and the trains are barreling toward each other on a collision path, at what time will they collide? I have to write an equation and solve.

Answer 1

Hey Liz,

 

First, we want to take note of all the information we're given, and consider what the question wants us to do... We're asked to find the "time of convergence" - when the two trains meet. If you look closely, then you'll notice that the trains don't leave at the same time of day, so we want 'to fix that' before we proceed any further. Below is the initial diagram we'll be using to start solving this problem:

 

----->                                                                 <-----

Chicago - - - - - - - - - - - - - - - - - - - - - - - - - Cleveland

 (6 PM)                       285 miles                        (7 PM)

 

In order to set the times of departure equal to one another, we simply re-evaluate the distance between the two cities @ 7 PM, and to do this we must see how far the train from Chicago travels after an hour (6 PM + 1 hour = 7 PM).

 

Since the Chicago train is travelling at 45 miles per hour, and the time of travel is 1 hour, we can easily solve and see that the Chicago train moves east 45 miles between 6 PM and 7 PM... Our new diagram is below:

 

           ----->                                                      <-----
           Chicago - - - - - - - - - - - - - - - - - - - - Cleveland
            (7 PM)                  240 miles                  (7 PM)

 

Now our distance to travel is 240 miles and our departure times are the same (7 PM)!

 

Okay, now to finish the solution to the problem, we set up our closing equation:

 

1) 45t + 35t = 240 ...

 

This first step mathematically represents the concept that, starting 240 miles away from one another, each train travels at its given velocity (v_Chicago = 45 MPH east, v_Cleveland = 35 MPH west) until collision.

 

2) 80t = 240

 

This second step simplifies the equation.

 

3) t = 3

 

This third step solves our equation.

 

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With t = 3, we know that it takes 3 hours from 7 PM for the trains to collide with one another, so the time of collision is

 

10 PM.

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I hope this explanation was helpful to you.

 

Best of luck with your studies.

 

 

 

Answer 2

Because the train from Chicago leaves 1 hour before the train from Cleveland, and because it travels at 45mph, we know that when the Cleveland train leaves, the Chicago train will have already traveled 45 miles. So, when the Cleveland train leaves, there will be 240 miles in between them. So, let's just assume they leave at the same time with 240 miles in between.

For these kinds of problems, we use the distance-rate-time formula:

 

             d = r*t

 

which simply states that how far something goes is equal to how fast it's going times how long it goes. 


For our problem,

let d represent the distance that the Chicago train will travel,

then the Cleveland train will travel (240 - d).

Let r₁ represent how fast the Chicago train is traveling.

Let r₂ represent how fast the Cleveland train in traveling. 

We know that, when they collide, both trains will have been traveling for the same amount of time (we made this true by adjusting the problem at the beginning when we assumed that the trains left at the same time). So, let t represent the amount of time the trains traveled upon collision. 

 

For our Chicago train, we now have the equation:

 

                 d = r₁*t                      (1)

 

For our Cleveland train, we now have the equation:

 

                 (240 - d) = r₂*t          (2)

 

Because we don't know what d is, we can substitute equation (1) into equation (2) and get rid of d.

 

                (240 - r₁*t) = r₂*t

 

We can now plug in our values for r₁ and r₂ and solve for t.

 

                (240 - (45)t) = (35)t

 

                240 - 45t = 35t                  (remove parentheses)

 

                240 = 80t                           (add 45t to both sides)

 

                t = 3.                                (divide both sides by 80)

 

So, they collide at 3 hours of travel. Given that they left at 7pm, the collision occurred at 10pm. 

-Ray Biggerstaff