Question

Two trains leave a railroad station at noon. The first train travels along a straight track at 90 mph. The second train travels at 75 mph along a straight track that makes an angle of 130 degrees with the first track. At what time are the trains 400 miles apart

Answer 1

Explanation:

Answer:

2:36pm

Explanation:

Let's say it takes x hours after leaving the station when the trains are 400 miles apart

Let's refer to the two trains using the letters A and B for easier reference

A is the train that travels on a straight track due East

B is the train that travels on a track 130° relative to the path of train A

A travels at a speed of 95 mph

B travels at a speed of 75 mph

Let the distance between the two trains be 400 miles after they have traveled an unknown hours. We have to determine

After hours, train A would be miles from the railroad station and train B would be miles from the station

We can visualize this situation as given in the attached image.

You can see that A = 95x, B = 75x and the longest side is 400 miles

We will use the law of cosines to determine x

The Law of Cosines (also called the Cosine Rule) says:

where is the angle opposite c i.e. the angle bounded by a and b which are the other two sides

Using the diagram as a reference we see

c = 400, a = 95x, b= 75x, 130°

Using this information and plugging values into the cosine rule formula we get

But we know that c = 400 since that's what the question says

So we get

So the two trains will be 400 miles apart 2.6 hours after they have left the station.

2.6 hours = 2 hours and 36 minutes

So the trains will be 400 miles 2 hours and 36 minutes after they have left the station

Since they have left at noon, they are 400 miles apart at 2:36pm

hope this helps ;)