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?

  • $\begingroup$ 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

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.