# Evaluate the maximum area of the rectangle touching the parabola.

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?

• 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. – NasuSama Sep 11 '17 at 0:24

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 $$A=w\times (a-\frac{w^2}{4})$$.
• 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. – Mathemagical Sep 11 '17 at 0:28