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CALCULUS

A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for $5 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $7000?

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  1. if the dimensions are x and y, then we have the cost function
    5*2x+2*2y = 7000
    5x+2y = 3500
    the area a = xy = x*(3500-5x)/2 = 1750x - 5/2 x^2
    da/dx = 1750 - 5x
    so maximum area is when the expensive sides are x = 350 ft
    and the cheap sides have length y = 875 ft

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