document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #359855 by KMST(5328) $\"\"$ $\"About$

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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.
\n" ); document.write( "The area fenced in is
\n" ); document.write( " $\"A=x%281000-2x%29=-2x%5E2%2B1000x\"$
\n" ); document.write( "The quadratic function
\n" ); document.write( " $\"A%28x%29=-2x%5E2%2B1000x\"$ , like all quadratic functions $\"f%28x%29=ax%5E2%2Bbx%2Bc\"$
\n" ); document.write( "with $\"a%3C0\"$ , has a maximum (and line of symmetry) at $\"x=-b%2F2a\"$ .
\n" ); document.write( "In this case, the maximum and line of symmetry are at $\"x=-1000%2F2%28-2%29=-1000%2F-4=250\"$ ,
\n" ); document.write( "So, 250 meters will be the width of the rectangular plot.
\n" ); document.write( "The length of the rectangular plot (along the river) will be (in meters)
\n" ); document.write( " $\"1000-2x=100-2%2A250=1000-500=500\"$
\n" ); document.write( "and the area (in square meters) will be
\n" ); document.write( " $\"A%28250%29=125000\"$ \n" ); document.write( "

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