two towers questions

[linse025]

1. The problem statement, all variables and given/known data
I need some help with this, actually quite a bit.

Two electric towers are 270 feet apart. The tower on the left is 75 feet tall the one on the right is 108 feet tall. A wire is strung between them that is tied to the ground. Where should the wire be placed so that it is the minimum length?


2. Relevant equations

The towers and the wire form two right triangles, so Pythagorean theorum must be used.


3. The attempt at a solution

http://img201.imageshack.us/img201/2294/towersml7.th.jpg (http://img201.imageshack.us/my.php?image=towersml7.jpg)

What I have come up with so far

total length of wire between tower A and the ground is X
total length of wire between tower B and the ground is 270-X

Total length of wire is the sum of the two wires
The formula I have so far is

http://img408.imageshack.us/img408/8162/equationhy8.th.jpg (http://img408.imageshack.us/my.php?image=equationhy8.jpg)

beyond ths step I have no idea what I should do

[robphy]

If you know calculus, you want to minimize Length by choosing a certain value of x.
If you know geometry, you have to think outside of the box.
If you know physics, you have to reflect on the problem a little more.

[linse025]

I've did some more looking, I'm taking this from the calculus approach.

The derivative of the function I have to find the length of the wire is:

http://img261.imageshack.us/img261/2492/equation2dp5.th.jpg (http://img261.imageshack.us/my.php?image=equation2dp5.jpg)

The next step is I set the derivative = 0, then solve for x

That's a pretty complex formula to solve so I plugged it into my ti-83 and it says that at x = 110.66 Y1 = 0

any ideas next?

[robphy]

So, aren't you done? You found x.
(By the way, you might try to plot the Length function to see that it truly is a minimum at your value of x.)

You might find it interesting to note that
110.66 / 270 = 0.409851852
75 / (75 + 108) = 0.409836066
which are essentially the same, up to round-off errors,
and similarly,
75 / 110.66 = 0.677751672
108 / (270 - 110.66) = 0.677795908.

If you plot your result on a diagram to scale [say on graph paper], you might be able to make sense of the above numerical similarities... as well as my hints above.