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18 July, 04:03

Optimization

a) An outdoor physical fitness complex consists of a rectangular region (football field) with a semicircle on each end. The perimeter of the complex is to be a 400-meter running track. What dimensions will create the maximum area of the rectangular section of the fitness complex?

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  1. 18 July, 04:09
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    length = 100 m

    width = 63.66 m

    Step-by-step explanation:

    Let's call

    x: lenght of rectangle

    y: width of rectangle, and also diameter of the semicircles

    Perimeter: 2*x + π*y = 400

    or:

    x = 200 - π/2*y

    Area of the rectangle = x*y

    Replacing:

    A = (200 - π/2*y) * y

    A = 200*y - π/2*y²

    At the maximum:

    dA/dy = 200 - π*y = 0

    y = 200/π

    y = 63.66 m

    and

    x = 200 - π/2*63.66

    x = 100 m
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