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  1. #1
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    Unhappy help pleaseeeeeeeee

    Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 7 cm and 8 cm if two sides of the rectangle lie along the legs.
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  2. #2
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    Hello, doneng!

    Find the area of the largest rectangle that can be inscribed in a right triangle
    with legs of lengths 7 cm and 8 cm if two sides of the rectangle lie along the legs.
    Code:
          |
        7 *
          |  *
          |     *
          |        * P(x,y)
          o - - - - - o
          |           |  *
          |          y|     *
          |           |        *
      - - o - - - - - o - - - - - * - -
          |     x                 8

    The equation of the hypotenuse is: . y \:=\:-\tfrac{7}{8}x + 7 .[1]

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

    Substitute [1] into [2]: . A \:=\:x\left(-\tfrac{7}{8}x+7\right) \quad\Rightarrow\quad A \;=\;-\tfrac{7}{8}x^2 + 7x

    Differentiate and equate to zero: . A' \;=\;-\tfrac{7}{4}t + 7 \:=\:0 \quad\Rightarrow\quad\boxed{ x \:=\:4}

    Substitute into [1]: . y \;=\;-\tfrac{7}{8}(4) + 7 \quad\Rightarrow\quad\boxed{ y \:=\:\tfrac{7}{2}}

    Therefore, the maximum area is: . A \;=\;xy \;=\;(4)\left(\tfrac{7}{2}\right) \;=\;\boxed{{\color{blue}14\text{ cm}^2}}

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