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 Math Central Quandaries & Queries
 Question from Ali, a student: Given an elliptical piece of cardboard defined by (x^2)/4 + (y^2)/4 = 1. How much of the cardboard is wasted after the largest rectangle (that can be inscribed inside the ellipse) is cut out?

Firstly, if you plot the "ellipse" you will see that it is a rather special ellipse, usually given another name.

Now there are two ways to proceed. The traditional way is to solve for one variable, say find y as a function of x. Then find the area as a function of $x: A(x) = x \times y(x),$ take its derivative, and set equal to 0.

You can also differentiate the constraint equation implicitly: $x/2 + (y/2) y' = 0$
differentiate the objective function (area) implicitly and set to 0: $y + x y' = 0$
and solve this system of equations:$y' = -y/x \rightarrow x^2 - y^2 = 0 \rightarrow x = \pm y$
then plug in to get the final solution.