x = width perpendicular to river
y = length parallel to river
fence length = 2 x + y = 32
x y = 128 so y = 128 / x
then
2 x + 128/x = 32
2 x^2 - 32 x + 128 = 0
x^2 - 16 x + 64 = 0
(x-8)(x-8) = 0
x = 8
then y = 128/x = 16
A farmer needs to enclose three sides of a field with a fence (the fourth side is a river). The farmer has 32 feet of fence and wants the field to have an area of 128 sq-feet. What should the dimensions of the field be? (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side). Additionally, the length should be as long as possible.)
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