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Show that the rectangle with the largest area that is inscribed within a circle of radius r is a square. Find the dimensions and the area of the inscribed square.

My respect goes to those who know how to tackle this one.

  • Calculus -

    Let the base of the rectangle be x, let its height be y units.

    the diagonal would be the diameter of the circle and it length is 2r.
    so x^2 + y^2 = 4r^2

    Area of rectangle = xy
    = x√(4r^2 - x^2)

    d(Area)/dx = ......
    = (4r^2 - 2x^2)/√(4r^2 - x^2)

    set this equal to zero for a maximum area and solve to get
    x = r√2

    put this back into x^2 + y^2 = 4r^2
    to get y = r√2

    so x=y, proving the rectange is a square

  • Calculus -

    Thank you so much. I appreciate you taking the time to answer my question. = )

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