The constraint for the maximum enclosed area is the amount the farmer is
willing to spend on fencing.
The width of the plot that will give the most area, y = <u>83.</u><u> feet</u>
The length of the plot that will give the most area, y = <u>125 feet</u>
Reasons:
The given parameters are;
The cost of fencing per foot = $20
The maximum amount the farmer is willing to spend = $5000
Number of sides of fencing = 3 sides
Cost of the west side of the = Split with neighbor
Solution:
Let, y, represent the length of the fence, and let <em>x</em> represent the width of
the fence, we have;
Length of fence required = y + 2·x
Cost of the fence = 20·y + 20·x + (20÷2)·x = 20·y + 30·x
Therefore;
Maximum amount to be spent on the fence, 5000 = 20·y + 30·x
Area of a rectangle = Length × Width
The enclosed rectangular area of the fencing, A = y × x
The leading coefficient of the quadratic function of the area is negative,
therefore, the function only has a maximum point.
At the point of the most area, the slope,
Finding the value of <em>x</em> at the maximum point, we get;
Therefore;
The length of the plot that will give the most area, y = <u>125 feet</u>
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