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Thread: Little optimization problem

  1. #1
    Junior Member
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    Exclamation Little optimization problem

    Hello friendsĦĦ

    I need to find the greatest area rectangle which can be drawn "inside" of a right triangle.

    Measures: Cathetus a = 4cm and cathetus b = 3cm.

    I need to solve the problem by using derivatives.

    Thanks in advance
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, osodud!

    Find the greatest area rectangle which can be drawn
    inside a right triangle with legs 3 and 4.
    A diagram will always help.
    Code:
    -   - *
    :   : |  *
    :  3-y|     *
    :   : |        *
    3   - *-----------*
    :   : |     x     |  *
    :   y |           |y    *
    :   : |           |        *
    -   - *-----------*-----------*
          : - - x - - :  - 4-x -  :
          : - - - - - 4 - - - - - :

    The area of the rectangle is: .$\displaystyle A \;=\;xy$ .[1]


    From the similar right triangles, we have: .$\displaystyle \frac{3-y}{x} \:=\:\frac{y}{4-x}$
    . . which simplifies to: .$\displaystyle y \;=\;3-\tfrac{3}{4}x$ .[2]


    Substitute [2] into [1]: .$\displaystyle A \;=\;x\left(3-\tfrac{3}{4}x\right)$


    And you must maximize: .$\displaystyle A \;=\;3x - \tfrac{3}{4}x^2$

    . . Got it?

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