The dimensions for the plot that would enclose the most area are a length and a width of 125 feet.
In this question we shall use the first and second derivative tests to determine the <em>optimal</em> dimensions of a rectangular plot of land. The perimeter (), in feet, and the area of the rectangular plot (
), in square feet, of land are described below:
(1)
(2)
Where:
- Width, in feet.
- Length, in feet.
In addition, the cost of fencing of the rectangular plot (), in monetary units, is:
(3)
Where is the fencing unit cost, in monetary units per foot.
Now we apply (2) and (3) in (1):
(4)
We notice that fencing costs are directly proportional to the area to be fenced. Let suppose that cost is the <em>maximum allowable </em>and we proceed to perform the first and second derivative tests:
FDT
SDT
Which means that length leads to a <em>maximum</em> area.
If we know that and
, then the dimensions of the rectangular plot of land are, respectively:
The dimensions for the plot that would enclose the most area are a length and a width of 125 feet.
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