SOLUTION: Juliet has 900 ft of fencing to build 6 animal pens. What dimensions for x and y maximize the total enclosed area? What is the total maximum area? What is the maximum size for one

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Question 969490: Juliet has 900 ft of fencing to build 6 animal pens. What dimensions for x and y maximize the total enclosed area? What is the total maximum area? What is the maximum size for one of the six animal pens?

The diagram shown 2x3 rectangle with six animal pens inside.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
If the pens measure x by y then the
perimeter of each is +2x+%2B+2y+
The perimeters of +6+ pens measure
+6%2A%28+2x+%2B+2y+%29+=+12%2A%28+x+%2B+y+%29+
+12%2A%28+x+%2B+y+%29+=+900+
+x+%2B+y+=+75+
+y+=+75+-+x+
------------------
+A+=+x%2Ay+
+A+=+x%2A%28+75+-+x+%29+
+A+=+-x%5E2+%2B+75x+
This is a parabola which is maximum when
+x%5Bmax%5D+=+-b%2F%282a%29+
+x%5Bmax%5D+=+-75%2F%282%2A%28-1%29+%29+
+x%5Bmax%5D+=+37.5+
and
+y+=+75+-+x+
+y+=+37.5+
---------------
To maximize area, the dimensions of
each pen are: 37.5 x 37.5
------------------------
Here is a plot of the function +A+=+-x%5E2+%2B+75x+
+graph%28+400%2C+400%2C+-10%2C+90%2C+-200%2C+1500%2C+-x%5E2+%2B+75x+%29+