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 s

Question

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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Amaya 1 day 2021-10-11T18:22:26+00:00 1 Answer 0

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    2021-10-11T18:24:01+00:00

    Answer:

    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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