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Math Help - Differentiation Applications

  1. #1
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    Differentiation Applications

    So I am having some trouble with a certian type of question that has arisen in some of the past work I have done leading me to believe that it could appear in my upcoming examination. So on with the question:

    A farmer plans to use a river as a one boundary of a rectangular paddock. If the farmer has 480m of fencing to be used to fence the remaining 3 sides, what dimensions should the paddock be to ensure maximum area.

    Help is greatly appriciated.
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  2. #2
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    Quote Originally Posted by holmesb View Post
    ...

    A farmer plans to use a river as a one boundary of a rectangular paddock. If the farmer has 480m of fencing to be used to fence the remaining 3 sides, what dimensions should the paddock be to ensure maximum area.
    Hello,

    let w be the width of the rectangle and
    l be the length of the rectangle.

    Then the area is calculated by:

    a = w \cdot l [1]

    the fence f consists of f = 2\cdot w + l. Since f = 480 you get:

    l = 480 - 2 \cdot w [2]

    Plug in the term for l into the equation [1]:

    a(w)=w \cdot (480-2 \cdot w) = -2\cdot w^2 + 480w

    This is a parabola opening downward, the maximum value is at it's vertex:

    a'(w) = -4\cdot w + 480 and a'(w) = 0

    You'll get: w = 120 m and l = 240 m. The maximum area is 28800 mē
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