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Given that a sports arena will have a 1400 meter perimeter and will have semi-
circles at the ends with a possible rectangular area between the semi-circles, determine
the dimensions of the rectangle and semi-circles that will maximize the total area.

  • math -

    Let the radius of the end sem-circles be r
    making the width of the rectangle to be 2r
    let the length of the rectangle be x

    The perimeter of the field is 2x + 2πr
    = 1400
    x + πr = 700
    x = 700-πr

    area = πr^2 + 2rx
    = πr^2 + 2r(700-πr)
    = πr^2 + 1400r - 2πr^2

    d(area)/dr = 2πr + 1400 - 4πr = 0 for a max of area
    1400 - 2πr = 0
    2πr = 1400
    r = 700/π

    then x = 700 - π(700/π) = 0

    Unexpected strange result!

    BUT there is nothing wrong with the calculations, it is the wording of the question that is flawed.
    Of course the largest area of any region with a given perimeter is a circle, which my solution has shown.

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