1. ## Optimization Problem

I really need help with this question pleaseee!!

A rectangle inscribed inside of an isosceles triangle whose hypotenuse is 2 units long. What is the largest are a the rectangle can have, and what are its dimensions?

2. Hello, cutiepie!

A rectangle inscribed inside of an isosceles right triangle with hypotenuse 2.
What is the largest area a the rectangle can have, and what are its dimensions?
I'll assume the diagram looks like this:
Code:
                      |
1*
* | *
*   |   *  P
* - - + - - o(x,y)
* |     |     | *
*   |     |     |y  *
*     |     |     |     *
- - * - - - + - - + - - + - - - * - -
-1          x  |  x          1

Point $P$ is on the line: $y \:=\:1-x$ .[1]

The area of the rectangle is: . $A \;=\;2xy$ .[2]

Substitute [1] into [2]: . $A \;=\;x(1-x) \quad\Rightarrow\quad A \;=\;x-x^2$

Maximize: . $A' \;=\;1 - 2x \:=\:0 \quad\Rightarrow\quad\boxed{ x \:=\:\tfrac{1}{2}}$

Substitute into [1]: . $y \;=\;1-\tfrac{1}{2} \quad\Rightarrow\quad\boxed{ y \:=\:\tfrac{1}{2}}$

Therefore, its dimensions are: . $\begin{array}{ccccc}\text{Length} &=&2x &= & 1\\ \text{Width} &=& y &=& \tfrac{1}{2} \end{array}$

. . and its maximum area is: . $A \;=\;(1)\left(\tfrac{1}{2}\right) \;=\;\tfrac{1}{2}$

3. Thank you Soroban. You just ended a night of complete and utter mathematical turmoil. You don't even know. We're indebted to your supreme awesomeness.