Example 2 Two
cars start out at the same point. One
car starts out driving north at 25 mph.
Two hours later the second car starts driving east at 20 mph. How long after the first car starts
traveling does it take for the two cars to be 300 miles apart?
Solution
We’ll start off by letting t be the amount of time that the first car, let’s call it car A,
travels. Since the second car, let’s
call that car B, starts out two hours later then we know that it will travel
for hours.
Now, we know that the distance traveled by an object (or
car since that’s what we’re dealing with here) is its speed times time
traveled. So we have the following
distances traveled for each car.
At this point a quick sketch of the situation is probably
in order so we can see just what is going on.
In the sketch we will assume that the two cars have traveled long
enough so that they are 300 miles apart.
So, we have a right triangle here. That means that we can use the Pythagorean
Theorem to say,
This is a quadratic equation, but it is going to need some
fairly heavy simplification before we can solve it so let’s do that.
Now, the coefficients here are quite large, but that is
just something that will happen fairly often with these problems so don’t
worry about that. Using the quadratic formula
(and simplifying that answer) gives,
Again, we have two solutions and we’re going to need to
determine which one is the correct one, so let’s convert them to decimals.
As with the previous example the negative answer just
doesn’t make any sense. So, it looks
like the car A traveled for 10.09998 hours when they were finally 300 miles
apart.
Also, even though the problem didn’t ask for it, the
second car will have traveled for 8.09998 hours before they are 300 miles
apart. Notice as well that this is NOT
the second solution without the negative this time, unlike the first example.
