Real world math question

So I was trudging a mile to my transmitter this morning. I got to thinking, about the quickest way to get from point a to point b is a straight line. Of course, I had pathagras' thearom in my head. We all learned about the 3-4-5 triangle. It is quickest to go the hypotenuse 5 rather than 3 and then 4.

But this does not take into account time.

Today, if I went a to b, it would have taken me 10 extra minutes because I would be walking through 1.5 feet of snow. However, if I went 3 and then 4, it was quicker....no snow.

Is there a theorem that takes into account time and difficulty of travel.

Thank you all.

What you are describing, is essentially the foundation of Calculus. If you've never studied Calculus before, pat yourself on the back, you have independently done something that took geniuses in previous centuries to comprehend.

The equation(s) you’re referencing aren’t multi-variable to account for other factors.

Once you figured out distance, you’d have to apply that to another equation that evaluates the speed at which you could move to figure out the fastest path.

Or, you could just know that it’s snowy and choose the less snowy route.

originally posted by: theatreboy
So I was trudging a mile to my transmitter this morning. I got to thinking, about the quickest way to get from point a to point b is a straight line. Of course, I had pathagras' thearom in my head. We all learned about the 3-4-5 triangle. It is quickest to go the hypotenuse 5 rather than 3 and then 4.

But this does not take into account time.

Today, if I went a to b, it would have taken me 10 extra minutes because I would be walking through 1.5 feet of snow. However, if I went 3 and then 4, it was quicker....no snow.

Is there a theorem that takes into account time and difficulty of travel.

Thank you all.

You are describing the affect of "the Coefficient of Friction".


Friction slowed you.

a reply to: theatreboy

Pythagorean's Theorem, A^2+B^2=C^2 for finding the hypotenuse of a right triangle (and most other formulas for triangles) are dealing with classic Euclidean geometry.....under ideal conditions.

Not really meant for real world problems which is what you're describing.

What you are describing is more of a physics problem because you are including new variables other then the length of 3 sides of a triangle.

You're including things like time, effort, energy, and force.

As others have stated, for many physics problems, you'll need to go from Algebra and simple geometry to Calculus in order to solve it.

On paper at least.

Another way to solve it is to simply perform the actions and see what your answers are.

a reply to: Archivalist

Guess it is time to get a Calculus book...I have 100 questions like these.

Also look into "Graph Theory". It is used extensively in Google Maps for plotting the fastest route for directions based upon your current position (which changes constantly).

Here's a neat explanation using a real world example: Discrete MathematicsAn Open Introduction: Chapter4 Graph Theory

a reply to: theatreboy

Yeah, if you make a corner of a triangle for two trains intersecting, you know the old trope.

One train travelling south one train travelling east at x miles per hour? That kinda question?

Well, those are set speeds, most of the time for algebraic math.

Calculus handles the same exact problem, but allows for you to factor in accelerations.

So you would have the trains having some arbitrary speed/velocity for an example, like 25 miles per hour, but then you throw in the monkey wrench. Accelerating at a rate of 6miles per hour every 13 seconds.

Calculus was made to solve for momentary points and/or any graphable point for those types of questions, when factoring in that the rate of speed is changing.

a reply to: Archivalist

So I just found the MIT Open Courseware with calc 18.01. This is going to be fun...thanks for pointing me in the right direction!

originally posted by: eriktheawful
a reply to: theatreboy

Pythagorean's Theorem, A^2+B^2=C^2 for finding the hypotenuse of a right triangle (and most other formulas for triangles) are dealing with classic Euclidean geometry.....under ideal conditions.

Not really meant for real world problems which is what you're describing.

What you are describing is more of a physics problem because you are including new variables other then the length of 3 sides of a triangle.

You're including things like time, effort, energy, and force.

As others have stated, for many physics problems, you'll need to go from Algebra and simple geometry to Calculus in order to solve it.

On paper at least.

Another way to solve it is to simply perform the actions and see what your answers are.

Congrats. You have just succinctly described the difference between theoretical physics and experimental physics.

What you are also describing is the fact that people are quite adept at doing applied math, physics, etc., on the fly with no clue that there may be proper equations or studies that sort it out in numbers. You could easily come up with a mathematical model for traversing terrain, snow and the like, but your mind is much quicker in many if not most cases.

Okay look at it this way. If I was laying in a field watching your quickest way avoiding the snow from right to left or vice versa I would see your traverse through a straight line. But then you go back and do it again the way you reckon is a straight line I would say no it isn’t.

my watching your detour from my viewpoint as a straight line I focus on the halfway point of the straight line route.
then the detour route is seen as a straight line. you are then focused on, on the straight line route. the line you deemed straight through your eyes.
how is the line straight? okay then let’s switch. is the line straight?
moving on as it could be said viewing point C from point A from the rear this is the question of timing being is point A to C quicker though? consider you decided to go from A to B to C in the snow from A to C in the same time and you check the time on your watch. if your watch is the same time as mine then where I am should be the same distance. you won’t even need to measure because it isn’t.

a reply to: theatreboy

I think the important thing to know about all the higher functions of mathematics, is that they are a way to arrive at a conclusion logically, IN THE ABSENCE of the intuition and instinct to arrive at those conclusions by more direct, less conscious means. So, for example, you COULD work out the shortest route, given all the possible variables, to arrive at point B, from point A in a given weather condition, using interesting applications of a host of different equations. But it would probably be a lot faster and a damned sight easier, to just let your instincts take over and come to the correct answer having done none of the work, and the reason I say that is because it is not only possible but very likely that your instinctive answer will be the correct one.

The same cannot be said of the things that using this sort of mathematics WILL be of benefit in, because the answers to said questions cannot be, or at least are much harder to arrive at instinctively, than they are logically. I suppose its a bit like this:

If you need a calculator and a pencil to work some things out, then you need more help than either item could ever provide you. Save the calculus for the stuff you could NEVER get right without it!