Another Nice Math Problem

YouTube is full of math videos now. That is a good thing if you know how to select the good ones. Unfortunately, there are numerous videos where math remains a mystery solvable only be geniuses. Some are just a recollection of achievements by some math hero of times long ago.

I believe that math should be entertaining too. Yes, it is demanding for the brain. But why not? And yes, you might not be able to solve any problem. Nobody is. Einstein was once quoted to have said: „To anybody who has troubles with math: Rest assured that my troubles have been bigger.“

I found the problem of this posting in this video by „DorFuchs“ (along with the unavoidable message from the sponsor NorthVPN).

The question is simply: We have a well with two sticks of 2m and 3m length leaning at the walls as shown, crossing at the water top at 1m. Can you determine the width of the well from the geometric configuration and the given values?

The geometrical equations are not easy to get if you are not used to that kind of problems. However, my first sketch is the following.

This is a very rough sketch, done with pen on my Asus. After trying for a while, I came up with the following equations.

By Pythagoras, we see

\(a^2+b^2=4, \quad c^2+b^2=9 \qquad \Longrightarrow \qquad c^2-a^2 = 5\)

Using similarity, we get

\(\dfrac{b}{a}=y, \quad \dfrac{b}{c}=x, \quad x+y=b \qquad\Longrightarrow\qquad \dfrac{1}{a}+\dfrac{1}{c} = 1\)

These are two equations for a and c. It looks as if the problem has at least locally a unique solution. However, it is involved to eliminate one variable. I did it automatically in Euler Math Toolbox.

>e1 &= c^2-a^2=5 with solve(1/a+1/c=1,a)
                            2      c
                           c  - -------- = 5
                                (c - 1)
>e2 &= ratsimp(e1)
                              4      3
                             c  - 2 c
                            ------------ = 5
                            c  - 2 c + 1
>csol = solve(&lhs(e2) with c=x,4,y=5)

Note, that I did the solution numerically. If it is done algebraically, Maxima yields a complicated expression involving third roots. The reason is that we are solving a fourth order polynomial. There are formulas for polynomials up to this order by Cardano. But for practical applications it is much easier to find the solution numerically.

Have a look:

I then tried to do a geometrical solution with my program C.a.R. Have a look at the following sketch.

We start the origin in the green dot. The blue dot is moveable on the x-axis. From this, we construct the points in the indicated order. In the proper solution, the red dot must fall onto the y-axis.

C.a.R. can generate automatic tracks of points while other points move on given lines. That has been done here. The track is clearly not a curve of second order, a circle or a parabola, or a line. Thus, it is no surprise that the problem cannot easily be solved algebraically.

It is possible to intersect such tracks with objects. That way we can find the value of c. However, the solution is only correct up to 5 digits due to the coarse way the track is generated.

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