Thread: application 2

problem: Two posts, one 3 meters high and the other 6 meters high, stand 10 meters apart. They are to be stayed by wires attached to a single stake at ground level, the wires running to the tops of the posts. Where should the stake be placed to use the least amount of wire?

Hello, cazimi!

This is one of the more unpleasant problems . . .

Two posts, one 3m high and the other 6m high, stand 10m apart.
They are to be stayed by wires attached to a single stake at ground level,
the wires running to the tops of the posts.
Where should the stake be placed to use the least amount of wire?

Code:

                                  *C
                                * |
                              *   |
                            *     |
     A*                   *       | 6
      |  *              *         |
    3 |     *         *           |
      |        *    *             |
     B* - - - - - * - - - - - - - *D
      :     x     P     10-x      :

AB is the 3-meter pole; CD is the 6-meter pole.
They are 10 meters apart: BD = 10.

The wire runs from A to a point P on the ground, then up to C.
Let x = BP, then PD = 10 - x.


From the right triangles (and Pythagorus), we have:
. . . . . . . . ______ . . . . . . . .___________
. . AP .= .√x² + 3², . CP .= .√(10 - x)² + 6²


The length of the wire is: .L .= .[x² + 9]^½ + [(10 - x)² + 36]^½

And that is the function you must minimize.

I'll wait in the car . . .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

There is a very clever geometric approach to this problem
. . which eliminates the need for all that Calculus.
I'll let someone else explain it.