MHB Solving a Challenging Math Problem | Expert Tips & Strategies

[math123456]

Hi All,

To better describe the problem I have uploaded here:

https://docs.google.com/document/d/13l9MZaSrdoh7oF7m75OB9jTRPnXlL_e7VXjUCFkqQBc/edit?usp=sharing

Feel free to ask any questions.

Any help is appreciated!

 

[HOI]

Take the original rectangle to have width "w" and height "h". Then the "reduced" ellipse has axe of length pw and ph, semi-axes pw/2 and ph/2, and is given by the equation \frac{x^2}{\left(\frac{pw}{2}\right)^2}+ \frac{y^2}{\left(\frac{ph}{2}\right)^2}= \frac{4x^2}{p^2w^2}+ \frac{4y^2}{p^2h^2}= 1.

Now, we have a rectangle Bwa by Bha that fits inside that reduced ellipse. Setting up a coordinate system with origin at the center of the ellipse, coordinate axes along the axes of the ellipse. In that coordinate system, the upper left corner of the rectangle has coordinates (Bwa/2, Bha/2) and, since that corner lies on the ellipse, must satisfy the equation of the ellipse. That is, we must have \frac{Bwa^2}{p^2w^2}+ \frac{Bha^2}{p^2h^2}= 1