Is there a mathematical equation or theorum which can depict the shortest route from A to B?

Question

For a company presentation I would like to state that the shortest route is best.Company guys are impressed by fancy geek like numbers .
Is there a scientific formulae that would take them apart -- by this I mean is there a proven method that the direct line from A to B is the most effective way of achieving the journey despite the aparant detour...Confused ...............?????

Answer 1

There is, but you have to state precisely what you want to show, for example:
- The shortest route is the best
- The shortest route between two points is the straight linear segment defined by the points.
- The shortest route (knowing that the direct line gives it) means the least time spent
- The shortest route (not knowing that the direct line gives it) means the least time spent
And you also have to decide how to solve it (algebraically/geometrically etc)
What do you mean by "aparant detour"
Problems that deal with degrees (bigger, biggest, smaller, less etc) are solved with Derivatives. Give an explanation of what you want to say and I'll try to answer.
And yes, the Triangle Inequality is a geometric proof, but you have to apply it correctly.

Answer 2

Your question kinda doesn't make sense...
despite WHAT apparent detour?

I will try to answer best i can but first you need to know the distance formula.

Here is the distance formula that works in the following way:

Imagine (or better yet DRAW) a right triangle where "d" is the hypotenuse of the triangle (the hypotenuse is the longest side, across from the right angle)

Label "x" as the bottom horizontal side and "y" as the other vertical side of the triangle.

Label the points A and B and C, where A and B are the endpoints of "d" and C is the point at the right angle

Now,

d= [sq (x^2 + y^2)]

You would say this in English as:
'd' equals the square root of 'x' squared plus 'y' squared.

Write this out on your paper as
(x squared plus y squared) under a big square root sign.

That equals the distance , d, between any two points, A and B.


Now to answer what i think you MAY be asking:
if the distance between any two points, A and B, in a triangle is the straight line between them, and distance is the SQUARE ROOT of the other two legs squared and summed,
then it would have to be SHORTER than adding up the ACTUAL values of the legs ( x and y ) and travelling the total distance of these from point A to get to B...

Hope that helps ;)

Answer 3

Yes, I'm confused. The shortest route is a straight line. How can a straight line be considered a detour?

Perhaps you're mixing up shortest and fastest.

Answer 4

Assume a number 'x',which is the length of the distance u wanted to travel from A to B,& let A be any number,say '1' to be easy. Now Ax=B,Keep changing x,if there is pt 'C' in between assume distance ac=x1,& distance cb=x2,& hence x becomes x1+x2,So keep changing the numbers for X ,& at point where u gets the shortest number for B,is the shortest route,Which will obviously be the straight route.

Hope i dint make it confusing...

Answer 5

The shortest route is a straight line. Or A -> B.

Answer 6

I'm not sure what exactly you want, but you can use the Triangle Inequality Thm.
Any other path would obviously pass a point C not on line AB. Then, you will have a triangle ABC. We know that AB< AC+CB by the Triangle Inequality Theorem. Any other path is longer therefore.

Answer 7

It's more just common sense. You could do proofs that A to B is quicker than say... A to C to B, but I do not believe there is a proof that would show that a line is the quickest way in EVERY situation.

It's just common sense.

Answer 8

one side of a triangle is always shorter than the sum of other two sides