SOLUTION: A petting zoo is working to construct a rectangular pen. The pen will have 1 partition to create separate spaces for goats and pigs. The petting zoo has 1000 feet of fencing to use

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Question 1085521: A petting zoo is working to construct a rectangular pen. The pen will have 1 partition to create separate spaces for goats and pigs. The petting zoo has 1000 feet of fencing to use.
a. Write an area function (A(x)) that gives the area of the enclosure as a function of the length of a partition, x.
b. Find the dimensions of the pen that will provide the greatest area.
c. What is the maximum area that can be enclosed?
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
x, a dimension of the whole pen
y, other dimension of the whole pen

Total fence length 1000 feet, and choosing x as the length of the one partition piece, for making two enclosures,
system%283x%2B2y=1000%2CA=xy%29

2y=-3x%2B1000
y=%281000-3x%29%2F2

A=x%281000-3x%29%2F2
highlight_green%28A%28x%29=%281%2F2%29x%281000-3x%29%29

A(x) would be a parabolic function, with vertex as a maximum. This x value for maximum will occur in the exact middle of the two zeros of A.
%281%2F2%29x%281000-3x%29=0, to find the two zeros.
x%281000-3x%29=0
1000-3x=0
1000=3x
x=1000%2F3=333%261%2F3
and the other zero is
x=0.

Maximum A will be at x=1000%2F%282%2A3%29=highlight_green%28166%262%2F3%29, feet.