Calculus

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What is the area of the largest rectangle that can be placed in a 5-12-13 right triangle (as shown)?

  • Calculus -

    Let ABC be the vertices of the triangle, right-angled at B, AB=5, BC=12 (vertical side), AC=13.
    Draw a rectangle BDEF, where D is on AB, E is on AC and F is on BC.
    Denote
    x=DE= height of rectangle
    Width of rectangle = DB = 5-(5x/12)
    Area of rectangle,
    A(x)=x(5-(5x/12))=5x-5x²/12
    A'(x) = 5-10x/12
    For A(x) to be maximum,
    A'(x) = 0 = 5-10x/12
    x=6, 5-5(6)/12 = 2.5
    The maximum area is 6*2.5=15
    Note that the aspect ratio of the rectangle is the same as that of the right sides of the triangle.

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