SOLUTION: A farmer with 1000 meters of fencing wants to enclose a rectangular plot that borders along a straight river. If the farmer does not want to fence along the river, what is the larg

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Question 551427: A farmer with 1000 meters of fencing wants to enclose a rectangular plot that borders along a straight river. If the farmer does not want to fence along the river, what is the largest area that can be enclosed? What dimensions produce that area?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The fence will go a distance x perpendicularly away from the banks of the river at two places, with a stretch of fence in between, parallel to the river, and measuring 1000-2x meters.
The area fenced in is
A=x%281000-2x%29=-2x%5E2%2B1000x
The quadratic function
A%28x%29=-2x%5E2%2B1000x , like all quadratic functions f%28x%29=ax%5E2%2Bbx%2Bc
with a%3C0, has a maximum (and line of symmetry) at x=-b%2F2a.
In this case, the maximum and line of symmetry are at x=-1000%2F2%28-2%29=-1000%2F-4=250,
So, 250 meters will be the width of the rectangular plot.
The length of the rectangular plot (along the river) will be (in meters)
1000-2x=100-2%2A250=1000-500=500
and the area (in square meters) will be
A%28250%29=125000