Construction Time

Geometry Level 4

In a construction project, Joe decides to stick up $5$ vertical poles, of lengths $1$ , $3$ , $5$ , $7$ , and $9$ . Each pair of consecutive poles have diagonal strands attached from the top of one pole to the bottom of the other. From the intersection of these diagonal strands, more poles are vertically stuck up. Using only the new poles, this process is repeated until one pole remains. To the nearest thousandth, find the length of this pole.

This problem is part of Tristan's set Formulas and Theorems .

The answer is 0.237.

2 solutions

Andrew Delgadillo
Apr 17, 2014

If you draw a diagram with line segments of lengths 1,3,5,7,9 perpendicular to the ground and draw diagonals from the top to bottom of the consecutive pole, you create a set of similar triangles. Solving for the height of the intersection above the ground reveals that the height above the ground is always (product of heights)/(sum of heights). Repeating this process with the new poles created in the previous run until you get the height of the last two poles (35/96, 315/464) you find the answer is 315/1328

Shashvat Shukla
Apr 23, 2014

The height of the intersection(h), can be expressed in terms of the height of the two poles(k and m) as $\frac{1}{h} = \frac{1}{m}+\frac{1}{k}$ Note that the distance between the poles does not matter. To prove this, coordinate geometry can be used. Using the coordinates of the tops and bottoms of the two poles, equations for the two diagonals can be found and by making x the subject and equating them, the height of the intersection can be found.

Now one can simply use this formula repeatedly to find the answer.

Construction Time

The answer is 0.237.

2 solutions

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