derive the ladder curve without calculus

In this video, I build intuition with limits to derive the bounding envelope of a falling ladder. I assume the ladder is sufficiently thin. I also assume knowledge of high school geometry and some trigonometry. Here we make use of algebraic manipulation, including fractional exponents.

Although the word "calculus" appears several times in my narration, this video is labeled "without calculus" in the sense that we are not doing any derivatives or integrals, nor setting up a difference quotient, as such--though a case can be made that we are doing calculus in disguise. Either way, the point is to help build intuition and confidence in the notion of limits, and negligible quantities, and to find the envelope equation.

I chose to use simple trig definitions repeatedly rather than a more elegant method. For example, the perpendicular lengths could be avoided by applying the Law of Cosines. The alternative and elegant approach to this problem, involves derivatives of parametric curves, while setting the ladder to unit length. I suppose I could have chosen to label (x_ell,y_ell) simply as (x,y) but here we are.

I recommend also watching my string art envelope video linked here:
   • the real string art symmetry and para...  

Special thanks to Patrick Sandiland for assistance with drawing the hybrid curve. Thank you JoAnne McKenny for background and Scott Seward and Family for help with audio.
Intro music was written and performed by me, Sobermath. All rights reserved.