Solving A Problem Involving Pythagorean Theorem And Polynomials

Question

I don't know how to solve this problem. What I've done so far is to construct 3 equations. However, I don't know how to solve those 3 equations:

Let y be the distance between the box and the ladder, and z be the length of the portion of the ladder that is beneath the top of the box:

$(x-1)^2+1=(10-z)^2$

$(y)^2+1=z^2$

$(x)^2+(1+y)^2=(10)^2$

I don't know how to proceed from here, however.

I know I solved this problem a few years back (possibly with different box size and ladder length) by setting up a fourth degree equation $f(x)=0$ such that $f(x)/x^2=0$ was a quadratic equation in $x+\frac1x$, but I can't seem to find that equation today. — Arthur, Oct 13 '18 at 09:42
See also: The position of a ladder leaning against a wall and touching a box under it. — Martin Sleziak, Oct 13 '18 at 13:54

Michael Rozenberg · Accepted Answer · 2018-10-13T07:56:16.177

6

By Pythagoras and similarity we obtain: $$\frac{x}{x-1}=\frac{\sqrt{100-x^2}}{1}$$ or $$x^4-2x^3-98x^2+200x-100=0,$$ where $1<x<10,$ or $$(x^2-x+1)^2-101(x-1)^2=0,$$ which after factoring gives: $$x=\frac{1}{2}(1+\sqrt{101}-\sqrt{98-2\sqrt{101}})$$ or $$x=\frac{1}{2}(1+\sqrt{101}+\sqrt{98-2\sqrt{101}}).$$

edited Oct 13 '18 at 07:56

answered Oct 13 '18 at 07:32

Michael Rozenberg

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Solving A Problem Involving Pythagorean Theorem And Polynomials

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