Question 549019: A farmer wants to divide his plot of land into 4 equal parts. He then will fence it in with 750 feet of fencing. Find a function that models the total area of the four pens.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
You can't do this problem exactly as it is stated. That is because there is no guarantee as to the shape of the plot of land.
Proceeding on the assumption that your problem should have read: "A farmer wants to divide his rectangular plot of land into 4 equal parts..."
Draw a rectangle. Construct a line segment with endpoints at the midpoints of the long sides of the rectangle. Construct a line segment with endpoints at the midpoints of the short sides of the rectangle. In sum, you will have 6 line segments that each measure one-half of the length of your rectangle and an additional 6 line segments that each measure one-half of the width of the rectangle. Therefore the total measure of the the line segments is  , and we are given that this total measure must be 750 feet, so:
which is to say
which is to say
Since the area of the original rectangle is given by  , then your Area (as a function of the width of the rectangle) would be:
\ =\ 250w\ -\ w^2)
Of course, all bets are off if the original plot was in the shape of a parallelogram, trapezoid, or some other odd-ball quadrilateral, or an ellipse, or...?
The area of a square field so divided and fenced would be exactly:
 square feet.
Which, by the way, is the maximum possible area for any rectangular shaped field.
The area of a circular field so divided and fenced would be exactly:
^2)
Which is approximately
 square feet
and any other regular polygon (except an equilateral triangle) would have an area somewhere between those limits.
John
My calculator said it, I believe it, that settles it
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