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| Math Central | Quandaries & Queries |
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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.
This method, while often easier, is not found in many textbooks.
Good Hunting!
RD
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