SOLUTION: Suppose you had a long stretch of fencing with which to make a nice big playpen for your new puppy. If you wanted to give your puppy the biggest possible area to roam around in, w

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Question 761966: Suppose you had a long stretch of fencing with which to make a nice big playpen for your new puppy. If you wanted to give your puppy the biggest possible area to roam around in, what shape should the fence be? Support your answer with sufficient data
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
It must be a square shape
Let the sides = +W+ and +L+
Let +A+ = area
Let +P+ = perimeter
+A+=+W%2AL+
Suppose you have +100+ units of fencing.
The units can be anything.
Now I can say
+P+=+2L+%2B+2W+
+100+=+2L+%2B+2W+
+L+%2B+W+=+50+
+L+=+50+-+W+
By substitution:
+A+=+W%2A%28+50+-+W+%29+
+A+=+-W%5E2+%2B+50W+
If you plot +A+ on the vertical axis and
+W+ on the horizontal, this is a parabola
The maximum is at +W%28max%29+=+-b%2F2a+
where the equation has the form +A+=+a%2AW%5E2+%2B+b%2AW+
+W%28max%29+=+-50%2F2%2A%28-1%29+
+W%28max%29+=+25+
and, since
+L+=+50+-+W+
+L+=+50+-+25+
+L+=+25+
Both length and width are 25, so +A%28max%29+ is a square