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A farmer wishes to enclose a rectangular pen with area 100 square feet next to a road. The fence along the road is to be reinforced and costs $34 per foot. Fencing that coast $16 per foot can be used for the other three sides. What dimensions for the pen will minimize the cost to the farmer. What is the minimum cost?

  • calc -

    Area= R*W where road fence is R, W is the length perpendicular to the Road.

    Cost= 34R+16(R+2W)= 34Area/W+ 16(Area/W+2W)
    dCost/dw= -34*100/W^2+ 16(-100/W^2+2)=0

    0=-3400-1600+32W^2

    W= sqrt (5000/32)

    L= 100/W

    check my math.

  • calc -

    let there be length a and width b, with side a along the road.

    b = 100/a

    cost is a*34 + a*16 + 2*100/a * 16

    c = 50a + 200/a
    c' = 50 - 200/a^2

    c' = 0 when a = 2

    so, the minimum cost is 100 + 100 = 200

    a 2' wide pen? Is he housing gerbils?

  • calc -

    My bad - bobpursley is correct. The road length is 8, width is 12.5

    using my notation,

    c = 50a + 3200/a

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