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Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 4 cm and 6 cm if two sides of the rectangle lie along the legs.

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2 answers

  1. let the base of the rectangle be x
    let the height of the rectangle be y

    If you make a sketch you can see three similar triangles,
    by ratios:
    (6-x)/y = 6/4
    6y = 24 - 4x
    y = (12 - 2x)/3

    area = xy
    = x((12-2x)/3 = 4x - (2/3)x^2
    d(area)/dx = 4 - (4/3)x
    = 0 for a max of area
    (4/3)x = 4
    4x = 12
    x = 3, then y = (12-6)/3 = 2

    So the largest rectangle has an area of 6 units^2

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  2. 6 on bottom
    4 up
    w horizontal
    h up from 6 side
    then
    w/6 = (4-h)/4
    or
    h/4 = (6-w)/6 (same thing by similar triangles)

    Area = (1/2) w h = A
    A =(1/2) w (4 - 2 w/3) = 2 w - w^2/3
    dA/dw = 0 at max/min = 2 - 2 w/3
    w = 3
    then h = 4 - 2(3)/3 = 2

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