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Find the area A of the largest rectangle that can be inscribed in a right triangle withh legs of length 3cm and 4cm if two sides of the rectangle lie along the two legs of the triangle.


    Place the right angle at the origin and let the 3 cm length be along the x axis and the 4 cm length along the y axis.
    Let x be the distance of one corner from the origin along the x axis.
    The hypotenuse has the equation
    y = 4 - (4/3)x
    For the rectangle thus formed,
    A = x*y = 4x - (4/3)x^2
    This has a maximum when
    dA/dx = 4 - (8/3) x = 0
    x = (3/8)*4 = 1.5
    y = 4 - (4/3)(3/2) = 2
    A = 3 cm^2

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