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18 October, 20:28

If the farmer has 250 feet of fencing to create a rectangular pen, define a function f that expresses the area of the field (measured in square feet) as a function of the width of the side of the field w (measured in feet). What is the maximum area possible?

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  1. 19 October, 00:27
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    Maximum area possible

    f (max) = 3906,25 ft²

    Dimensions:

    a = 62,5 ft

    w = 62,5 ft

    Step-by-step explanation:

    Perimeter of the rectangular fencing P = 250 feet

    And sides of the rectangle a and w (width of rectangle)

    Then

    A = a*w

    2a + 2w = 250 ⇒ a = (250 - 2w) / 2 ⇒ a = 125 - w

    f (w) = (125 - w) * w f (w) = 125w - w²

    Taking derivatives both sides of the equation

    f' (w) = 125 - 2w f' (w) = 0 125 - 2w = 0

    w = 125/2

    w = 62,5 ft ⇒ a = 125 - 62,5

    a = 62,5 ft

    f (max) = (62,5) ²

    f (max) = 3906,25 ft²
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