Math as a Muse (Part II)

In my previous post, I described a math problem that kept my mind busy during a church service a couple of weeks ago.  For a description of the problem, the link is here.  In any case, I will show the image that describes the problem again below...

As people sang their hymns, I began to think of the trigonometry of the situation - of the right angle triangles that are formed.  By the end of that hymn, I realized that actually, the line of sight to the nth column (pew) forms two right angle triangles which were geometrically similar (same internal angles).  As a result, the ratio of the two triangles' opposite and adjacent sides was the same.

As the sermon began, I was busily trying to determine the values for those adjacent and opposite sides:

For triangle 1 (the bigger one), opposite = H, and adjacent = s(n-1)+L.
For triangle 2 (the smaller one), opposite = h(n), and adjacent = s.

Then, putting the ratios equal to each other, (opp/adj)1 = (opp/adj)2, I was able to solve for the general function, h(n), that I sought:

h(n) = Hs/[s(n-1)+L] or h(n) = H/[(n-1)+(L/s)]

So, by the end of the sermon, I had it...The best part about solving a problem like this is that you get to test it immediately.  For a function, one can do a parametric analysis, that is, check how changes in each variable affect the function.

Here, it is easy to see that the whole function is proportional to H.  This makes sense, because if H goes to zero, I see no columns other than the zeroth one.  I could then stand on my tippy toes, thereby doubling H, and see that indeed, my view of all of the other pews doubled as well.

This function agreed also with my initial observation, that my view of subsequent columns decreases as n goes up.  One interesting surprise, however, was the link between L and s.  Their ratio actually shifts the function to the left.  I could test this as well: when I doubled L, by taking a small step backwards, I saw slightly less of each pew.  Of course, I could not adjust the pew spacing (s), but I could predict the following: dividing s by two affected h(n) in precisely the same way as multiplying L by two.  This was particularly neat, because it meant that I could envision just how much more or less of each pew I would see if they were spaced closer or further apart by making slight adjustments to where I stood.

I have plotted h(n) below for reasonable values of the other parameters (s = 0.8 m, H = 0.3 m, L = 0.4 m):

Not surprisingly, h(n) is always decreasing - the visible height approaches zero as n approaches infinity.  This kind of a function is known as a rational function.

A religious institution may seem like an odd place to perform mental gymnastics of the mathematical variety, but truly, I cannot think of anything more pure and divine than geometry.