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area of a rectangular region: a farmer wishes to create two rectangular regions bordering a river, by three fences perpendicular to the river and one connecting them. suppose that x represents the length of each of the three parallel pieces of fencing. she has 600 feet of fencing available.

a) what is the length of the remaining piece of fencing in terms of x?
b) determine a function A that represents the total area of the enclosed region.
c) give any restrictions on x
d) what dimensions of the total enclosed region would give an area of 22,500 feet squared?
e) what is the maximum area that can be enclosed?

  • pre-calc -

    600 feet total fence
    3 sides of x feet

    remaining side: 600 - 3x

    a(x) = x(600-3x) = 3x(200-x)

    naturally, 3x < 600, so x < 200 assuming an infinitesimally thin fence and poles of zero diameter. :-)

    22500 = 3x(200-x)
    -3x^2 + 600x - 22500 = 0
    x = 50 or 150

    max area achieved at x = 100
    a(100) = 30,000

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