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21 April, 20:34

A farmer decides to build a fence to enclose a rectangular field in which he will plant a crop. he has 1000 feet of fence to use and his goal is to maximize the area of his field.

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  1. 21 April, 21:04
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    So a rectangles area is a*b (two sides), its perimater is 2a+2b

    We know that 2a+2b=1000 or a+b=500

    Nowing this we need to find the max value for a*b

    a*b=a * (500-a) = 500a - a^2 = - (a^2-500a)

    For simplicity let us work with only a^2-500a

    To find the minimum or maximum of a parabola you need to create a perfect square. (like this: (x+y) ^2 - C where C is a constant)

    a^2-500a = (a-250) ^2 - 250^2

    So this - (a^2-500a)

    becomes:

    - (a-250) ^2 + 250^2 and you would like to find the max value.

    The first part - (a-250) ^2 can be 0 or negative so the max value will be when it is 0.

    Thus a=250 - > b=250

    This is no big surprise as with given perimeter the biggest area of a rectangal we can get is a square.
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