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muckyMuckMan
07-06-2002, 05:40 PM
Hi, I was just reviewing some calc problems and came across this one. Can anyone help me understand how this problem is to be solved? (Hints, tips, or whatever)
Minimum Length:
Two factories are located at the coordinates (-x, 0) and (x, 0) with their power supply located at the point (0, h). Find y such that the total amount of power line from the power supply to the factories is a minimum.

Here's a picture:
http://home.earthlink.net/~funbox/math/opprob.gif

muckyMuckMan
07-11-2002, 07:54 PM
Ah!! Duh, I finally got it. It ends up I was doing it right all along. I just was doing some funky algebra at the end. Here's how I did it if anyone's interested. Hehe, I also made a Flash demo so I could see what was going on.


Let L = total length of the power line.
h and (+/-)x are constants.

L = (h - y) + 2*sqrt(x^2 + y^2)
L' = -1 + 2y
-------------
sqrt(x^2 + y^2)

Now set L' = 0


0 = -1 + 2y
-------------
sqrt(x^2 + y^2)

1 = 2y
-------------
sqrt(x^2 + y^2)

sqrt(x^2 + y^2) = 2y

(sqrt(x^2 + y^2))^2 = (2y)^2

x^2 + y^2 = 4y^2

x^2 = 3y^2

x^2 = y^2
---
3

x = y
------
sqrt(3)

Yay!