Solve Two Towers Question: Minimum Length of Wire

  • Thread starter linse025
  • Start date
In summary, the wire should be placed between the towers such that the wire is the minimum length, given that the towers are right triangles and the wire forms a Pythagorean theorem.
  • #1
linse025
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Homework Statement


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?


Homework Equations



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


The Attempt at a Solution



http://img201.imageshack.us/img201/2294/towersml7.th.jpg [Broken]

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 [Broken]

beyond ths step I have no idea what I should do
 
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  • #2
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.
 
  • #3
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 [Broken]

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?
 
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  • #4
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.
 

1. What is the Two Towers Question?

The Two Towers Question is a mathematical problem that involves finding the minimum length of wire needed to connect two towers of different heights. The towers are placed on a flat surface and the wire must start from the top of one tower, go down to the ground, and then go up to the top of the other tower.

2. How is the minimum length of wire calculated?

The minimum length of wire is calculated by using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the wire) is equal to the sum of the squares of the other two sides (the height of each tower). By finding the square root of this sum, we can determine the minimum length of wire needed.

3. Is there a specific formula to solve this question?

Yes, the formula to solve the Two Towers Question is the square root of (h12 + h22), where h1 and h2 are the heights of the two towers in feet. This will give the minimum length of wire needed in feet.

4. Can this question be solved without using the Pythagorean theorem?

Yes, there are other mathematical methods that can be used to solve this question, such as using trigonometric functions or using the law of cosines. However, the Pythagorean theorem is the most straightforward and commonly used method for solving this problem.

5. What factors can affect the minimum length of wire needed?

The only factors that can affect the minimum length of wire needed are the heights of the two towers. The distance between the towers, the angle of the wire, and any other external factors do not affect the minimum length of wire required.

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