My name is Chris Baranski and I have a math problem. I am in grade 10 but I think the question is far beyond grade 10 level. My dad got this problem while attending university and since I enjoy math he gave me this problem which he has never found an answer to. Here it is:
I don't have a nice way to solve this problem. The solution given here rersults in a quartic which I then solve by approximation.
In the diagram below |AC| = 3 and |BD| = 2 and |BC| = x is the width of the well.
It is more convienent to work with y = |CD|, but Pythagoras' Theorem for triangle DBC gives
Likewise Pythagoras' Theorem for triangle ABC gives
Triangles DEC and ABE are similar anf thus,
Also triangles DEF and BEG are similar and hence
Squaring both sides yields,
At this point I used some graphic software to plot the graph of f(y) = y4 -2y3 + 5(y-1)2 to see where it crosses the x-axis. It does so twice at y = 0.7 and y = 1.58. Since y must be at least 1 the correct solution is y = 1.58 and since x2 + y2 = 4, x = 1.23.