Problema Solution
James has 900 yards of fencing and wishes to enclose a rectangular area. This rectangular are has one of the long sides against a barn and thus not requiring fencing. Express the area A of the rectangle as a function of the width x of the rectangle. For what value of x is the area the largest? What is the maximum area?
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x = the width of the rectangle
y = the length of the rectangle
This rectangular are has one of the long sides against a barn and thus not requiring fencing mean:
2x + y = 900
y = 900 - 2x
A = width*length is the area
A = x*y
A = x(900 - 2x)
A = -2x^2 + 900x
we need to find the maximum of A = -2x^2 + 900x
A max = c - b^2/(4a), where a = -2, b = 900, c = 0
A max = 0 - 900^2/(4*(-2))
A max = 101,250 yd^2 is the maximum area
by solving -2x^2 + 900x = 101250 we find
x = 225 yd
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for x = 225 yd the area is the largest.