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If I have a rectangle inside a parabola with the equation $y = a - x^2$, such that:

  • $a \geq 0$
  • Its base lies on the $x$-axis
  • Two of its vertices touch the parabola

What is maximum area of the rectangle?

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  • $\begingroup$ I have included the extra condition $a \geq 0$ for this problem since the maximum area is finite for that interval. Here is the related problem. $\endgroup$ – NasuSama Sep 11 '17 at 0:24
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The area of the rectangle is $A=hw$. But $h$ depends on $w$, $w/2$ is the x-distance from the origin ($w$ represents width) so $h=a-(\frac{w}{2})^2 $.

Now we have to maximise $A$ in $$A=w\times (a-\frac{w^2}{4})$$.

Can you do that? Hint: find the roots of the derivative.

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  • $\begingroup$ Another hint: The answers are different for $a<1$ and $a>1$. In a sense, the solution (of a rectangle above the x-axis) does not exist for one of the cases. $\endgroup$ – Mathemagical Sep 11 '17 at 0:28

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