# Thread: Square in A Parabola

1. ## Square in A Parabola

A rectangle PQRS is inscribed as sketched in the region between the X axis and the part of the graph y= cos(4x) specified by -(8/Pi) less then or equal to x less then or equal to (8/pi). Determine the coordinates of P for which the perimeter of PQRS is a maximum.

2. Hello, Rimas!

A rectangle $\displaystyle PQRS$ is inscribed as sketched in the region between the [tex]x[/mathy]-axis
and the part of the graph $\displaystyle y\:=\:\cos(4x)$, on $\displaystyle [-8\pi,\,8\pi]$
Determine the coordinates of $\displaystyle P$ for which the perimeter of $\displaystyle PQRS$ is a maximum.
Code:
                |
*
*   |   *
*- - - | - - -*
* |      |      | *
*  |y     |     y|  *
|      |      |
---*---+------+------+---*---
-8π       x  |  x      8π

The perimeter of the rectangle is: .$\displaystyle P \:=\:4x + 2y$ .where $\displaystyle y \:=\:\cos(4x)$

So we have: .$\displaystyle P \:=\:4x + 2\cos(4x)$

Set the derivative equal to zero: .$\displaystyle P\:\!' \:=\:4 - 8\sin(4x) \:=\:0 \quad\Rightarrow\quad \sin(4x) \,=\,\frac{1}{2}$

. . Hence: .$\displaystyle 4x \:=\:\frac{\pi}{6} \quad\Rightarrow\quad x \:=\:\frac{\pi}{24}$

. . Then: .$\displaystyle y \:=\:\cos\left(4\cdot\frac{\pi}{24}\right) \:=\:\cos\left(\frac{\pi}{6}\right) \:=\:\frac{\sqrt{3}}{2}$

Therefore: .$\displaystyle P\left(\frac{\pi}{24},\:\frac{\sqrt{3}}{2}\right)$

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### square inscribed in a parabola

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