How long after the first car starts traveling does it take for the two cars to be 300 miles apart?

Question

Two cars start out at the same point.  One car starts out driving north at 30 mph.  Two hours later the second car starts driving east at 25 mph.  How long after the first car starts traveling does it take for the two cars to be 300 miles apart?

 

Answer 1

Let the car starts from P. So when the second car starts the car A is already 60 miles away from P.

Then the car B starts towards east at 25mph and let the time taken to reach 300 miles apart be t.

Using pythagoras theorem and distance=speed*time we can formulate the given problem as:

300^2 = (60+30t)^2 +(25t)^2

or 90000 = 3600 + 3600t + 900t^2 + 625t^2

or 1525t^2 +3600t -86400 =0

on solving the above equation using quadratic formula we get:

t = -8.8 or t = 6.4

neglecting negative value for t as time cannot be negative.

So it will take roughly 6.4+2=8.4 hours for both the cars to get 300 miles apart from the initial time.

Answer 2

The first car would have already travelled 60 miles north when the second car sets off.

After time t hours the first car would travel an additional 30t miles north while the second car would travel 25t miles east. The total distance for the first car is 60+30t, and the hypotenuse of a right triangle is their distance apart=√((60+30t)²+(25t)²)=300, 3600+3600t+900t²+625t²=90000.

1525t²+3600t=86400, 61t²+144t-3456=0. Using the quadratic formula, t=6.4387 hrs, (6hr 26.32min approx). This is the positive root. But t is time measured by the second car, so we have to add two hours to this: 8.4387hr or 8hr 26.32min.