SOLUTION: A farmer needs to build a fence with 2 equal size pens to separate the sheep and pigs. The farmer will build a rectangular outer fence plus a dividing fence down the middle paralle

Algebra ->  Customizable Word Problem Solvers  -> Misc -> SOLUTION: A farmer needs to build a fence with 2 equal size pens to separate the sheep and pigs. The farmer will build a rectangular outer fence plus a dividing fence down the middle paralle      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 760995: A farmer needs to build a fence with 2 equal size pens to separate the sheep and pigs. The farmer will build a rectangular outer fence plus a dividing fence down the middle parallel to two sides. If she has a total of 200 feet of fencing, find the dimensions that will enclose the maximum total area.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x for length of the whole area, and y for the width of the whole area. Choose y longer than x, and so take the divider parallel to x.

Accounting for fencing material, 3x%2B2y=200.
Accounting for area, A=xy

Perimeter equation gives 2y=200-3x
y=100-%283%2F2%29x
Substitute into area equation.
A=x%28100-%283%2F2%29x%29
A=100x-%283%2F2%29x%5E2

A looks like a quadratic equation as a function, and indicates a parabola with a maximum ( openings downward). Finding the zeros is very easy, only requiring back one step in deriving A(x). The maximum A for AREA will occur directly in the middle of the two zeros of A(x).

ZEROS of A:
0=x%28100-%283%2F2%29x%29
Either x=0
OR
100-(3/2)x=0
100=3x%2F2
100%2A2%2F3=x
x=200%2F3
Maximum Area occurs at %280%2B200%2F3%29%2F2=highlight%28100%2F3=x%29
To find y,
y=100-%283%2F2%29x
y=100-%283%2F2%29%28100%2F3%29=100-3%2A100%2F%282%2A3%29=100-100%2F2
highlight%28y=50%29


----------------------------------More about the area-----------
This area maximum is:
A(100/3)=100(100/3)-(3/2)(100/3)^2
A=10000/3-(3/2)*10000/9
A=1000/3-3*10000/18
A=6*10000/18-3*10000/18
A=3*10000/(3*6)=10000/6
A=1667 square feet