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23 June, 06:53

Carol has 800 ft of fencing to fence in a rectangular horse corral which is bordered by a barn on one side. find the dimensions that maximizes the area of the corral.

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  1. 23 June, 07:24
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    200 ft * 400 ft

    Step-by-step explanation:

    Let x = one dimension of the corral

    and y = the other dimension

    Carol is using the barn on one side, so she needs to fence in only three sides (as in the diagram below).

    Then

    (1) 2x + y = 800 (Formula for perimeter)

    (2) y = 800 - 2x Subtracted 2x from each side

    (3) A = xy (Formula for area)

    A = x (800 - 2x) Substituted (2) into (3)

    A = 800x - 2x² Distributed the x

    This is the equation for a downward-opening parabola.

    The vertex (maximum) occurs at

    x = - b / (2a), where

    b = 800 and

    a = - 2

    x = - 800/[2 (-2) ] = - 800 / (-4)

    (4) x = 200 ft

    This is the value of x that gives the maximum area.

    2*200 + y = 800 Substituted (4) into (1)

    400 + y = 800

    y = 400 ft Subtracted 400 from each side

    This is the value of y that gives the maximum area.

    The dimensions that maximize the area of the corral are 200 ft * 400 ft.

    The graph of A = 800x - 2x² shows that the area is a maximum

    when x = 200 ft.
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