The View from The Window

What a great title for a love poem. This is not a poem, at least not in the traditional sense of the word.
From the window in the NHTI Math Lab where I tutor, a cement pathway with a metal railing can be seen. While (not) tutoring one particularly boring day, I noticed that the metal was held up with vertical metal posts spaced evenly apart. It looked something like this:

As I looked at this railing I noticed that some of the vertical parts lined up in my line of sight, and some didn’t. Curious, I drew a bird’s eye view picture:

I realized that I could count the number of spaces between the lined up posts in the back and in the front (in the case of the picture, four and three respectively). Using similar triangles gives the ratio:

Where the number of spaces in the front is , the number of spaces in the back is , the distance between me and the nearest railing is and the width of the walkway (the distance between railings) is . This ratio can be rearranged as:

Using this formula, and the values taken from the picture ( and ), it could be said that the distance from me to the nearest railing is four times that of the width of the walkway. This doesn’t say how far away the railing is, only what that distance is relative to the walkway width.

I really wanted to figure out if there was a way to tell how far the railing was without going outside. That’s when I realized that by taking a few steps backwards, different spokes lined up. I called the new number of spaces in the front , the new number of spaces in the back and the distance between where I was standing before and my new position , I could set up a new ratio:

This too can be rearranged:

Putting together both equations and isolating gives:

This formula gives the distance between me and the walkway in terms of the distance between two different viewing spots and observations made at each of those spots. Because both spots are inside the Math Lab, I never need to go outside to find that distance.

Naturally, I have no interest in ever actually finding a value for the distance, only in finding out if it could be done. It can, now I should go do something else.

-Nick

[This is a reprint of an essay I wrote for NHTI's annual publication, The Eye. These are not the original drawings because I couldn't figure out how to get them out of the Word file.]