SOLUTION: please help solve with solutions: two cars travels at a rate of 30 kph and 40 kph respectively at the same point in a right angle position. how long will they be 100 kms apart

Question 880779: please help solve with solutions: two cars travels at a rate of 30 kph and 40 kph respectively at the same point in a right angle position. how long will they be 100 kms apart

Answer by KMST(5347) (Show Source): You can put this solution on YOUR website!
The distances (in km) traveled by one car an the other form the legs of a right triangle.
The distance from one car to the other would be the hypotenuse.
The Pythagorean theorem says that the lengths of the legs and hypotenuse of a right triangle ( , , and )
are related by .
You are expected to write some equation using that and solve it.
It is good to know how to do that
However, if you are faced with this problem on a timed test, a mental math approach would be very efficient, and with that approach I knew that the answer is that it will take for the cars to be 100 kms apart.

THE EXPECTED WAY TO SOLVE IT:
= time in hours from the time the cars start traveling away from their common starting point.
= distance (in km) traveled by the car moving at a rate of 30 kph.
= distance (in km) traveled by the car moving at a rate of 40 kph.
Applying the Pythagorean theorem we get our equation:
.
Now, we simplify and solve:

That equation has the solutions and ,
but the way the problem asks the question, makes no sense, because you want a positive number for the answer,
so it will take for the cars to be 100 kms apart.
In reality, what the and answers tell you is that if two cars are moving at those speeds on perpendicular roads, and cross paths narrowly missing a collision, they will be 100 km apart 2 hours before and 2 hours after they cross paths.

THE MENTAL MATH TRICK:
Math problems are usually carefully designed to give results that are look nice and are easy to calculate. If possible, we like integer numbers.
There are some trios of natural numbers (positive integers) that fit the Pythagorean relation.
They are called Pythagorean triples, and you can search online and find lists of the first few. (I believe nobody listed the expected infinity of them).
The only one that all math teachers remember is (3,4,5).
(It is the only one I remember.
Because , a right triangle with legs measuring 3 and 4 of whatever unit will have a hypotenuse measuring 5 of the same unit.
Of course, you can scale up that 3-4-5 triangle to get one with sides measuring 6, 8, and 10, or 60, 80, and 100. (I can do that calculation in my head, can you?)
The problem asked for a hypotenuse measuring 100 km, and that meant legs measuring 60 km and 80 km.
For a car moving at 30 kph to travel 60 kn it would take . (I can do that calculation in my head, can you?)
Right triangles with side length in a 3:4:5 ratio show up in many problems, so it is a useful ratio to keep in mind.
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