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19 February, 01:05

A rancher has 1,800 linear feet of fencing and wants to enclose a rectangular field and then divide it into two equal pastures with one internal fence parallel to one of the rectangular sides. what is the maximum area of each pasture? 5400

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  1. 19 February, 01:27
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    Answer: 67.500 ft²

    Explanation:

    1) Name the dimensions using variables:

    y: length of the rectangular field

    x: widht of the rectangular field

    2) Model the amount of fence used by the two equal pastures:

    two sides and one internal fence: 2x + x = 3x

    two lengths: 2y

    ⇒ 3x + 2y = 1800 ← linear feet of fence

    y = 1800 / 2 - 3x/2 ← solving for y

    y = 900 - 3x/2

    3) Area of each pasture

    A = x (y/2) ← half ot xy

    A = x (900 - 3x/2) ← replacing y with 900 - 3x/2

    A = 900x - 3x² / 2 ← using distributive property

    4) Maximum area ⇒ A' = 0

    A' = 900 - 3x ← derivative of the polynomial 900x - 3x² / 2

    900 - 3x = 0

    ⇒ 3x = 900

    ⇒ x = 900/3

    ⇒ x = 300

    4) Determine y

    y = 900 - 3x / 2 = 900 - 3 (300) / 2 = 900 - 450 = 450

    5) Area of each pasture

    A = xy/2 = 300 * 450 / 2 = 67500 ← final answer
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