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Math Help - Optimization Problem

  1. #1
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    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?
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  2. #2
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    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}


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  3. #3
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    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.
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