Calculus Problem without Calculus

[tea217]

Picture

In the accompanying diagram, locate the point P so that the path APB has minimum length.
(In case the side lengths are not clear, they are 20m, 50m, 200m)

Sigh...i got the answer, which is 57 1/7 with calculus methods, but i need a NON-CALCULUS solution, and i donno how to do it without calculus.
ANyone??

 

[JonBarillari]

define c and d as the end points of the 200m line.

The minimum length occurs when triangle ACP is congruent with triangle BDP thus when: 20/x = 50/200-x

Hope this helps!

 

[her209]

What about an isoceles triangle?

 

[tea217]

I know about the isoceles triangle thing....and about the similar triangle thing.....
but...how can i prove that the MINIMUN occurs when we have similar triangle..??????

 

[killface]

Side of triangle A=x and B=2.5x (50/20 = 2.5) therefore 200=x+2.5X, X=57.14.

Edit - it's just using similar triangles - JonBarillari did the same thing. I think you're making it harder than it really is.

 

[tea217]

I KNOW THEY ARE SIMILAR TRIANGLES when the length AP to PB is minimun.

but WHY man???
i have to give a damn reason....or else i won't get the damn mark for it...

I thought for it like..for 5hrs...and still...i can't figure out why are the triangles similar when the lenght is minimun....

 

[911paramedic]

because the shortest distance is a straight line, since you cant have that, it needs to be a right angle (ap,pb,ba) where angle p is your right angle.

I think, lol

 

[JonBarillari]

Angle p is not necessarily a right angle, I haven't heard of the isoceles thing, nor of any other proof, I just came up with the congruent theory myself today.

My reasoning was: to minimize AP + PB, you must create angles at P for BOTH triangles formed as close to 45 degrees as possible, to do this, the optimal way is to create congruent triangles.

Does this make sense?!? Sorry that I don't have a more formal proof.

 

[killface]

nevermind

 

[tea217]

JonBarillari
Sorry...i don't get it.

....>.<

 

[tea217]

And also....i don't think you want to get the angles a P to get to 45 as close as possible.......
cuz..the actual angles when u are at min length is 19.3

 

[JonBarillari]

<< i don't think you want to get the angles a P to get to 45 as close as possible >>



I stand corrected, your right.

You want similar triangles because.... I am not sure how to explain it ... it just makes sense to me..

Sorry

 

[tea217]

sigh....my teacher told me that....the answer to this question is so simple... even a grade4 kid can understand it...

And the hint he gave us was that the shortest distance between 2point is a straight line....

 

[Noriaki]

Why preciesely do you want to do an optimaztion problem without calculus?

I don't know if there is a way to prove it without....

 

[JonBarillari]

<< the hint he gave us was that the shortest distance between 2point is a straight line.... >>



That's it!! Now I understand!

Flip one side and draw a line between them (does this make sense)? Let me try to draw what I mean:

I\
I_\___
.....\....I
.......\..I
.........\I

point p is the middle of the diagram... (the dots are place holders cause spaces don't work)

now you got your straight line and the construction of the similar triangles... then carry on with the agreed calculations.

 

[BigNeko]

JonB,
I was trying to remember that! You got it.

*passes JonB a cookie*

 

[tea217]

JonBarillari

I understand now.... thank you very much