document.write( "Question 1099767: a) A rectangular pen is build with one side against a barn. If 2100 m of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen?
\n" ); document.write( "b) A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 400m^2. What are the dimensions of each pen that minimize the amount of fence that must be used? \n" ); document.write( "

Algebra.Com's Answer #714244 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
I see the first question as a quadratic function question,
\n" ); document.write( "although you need to know enough geometry to understand what a rectangle is.
\n" ); document.write( "The second question, looks more like a calculus question
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\n" ); document.write( "a)

The area $\"y\"$ of that pen (in square meters) is $\"y=x%282100-2x%29\"$ .
\n" ); document.write( "You should recognize that as a quadratic function,
\n" ); document.write( "which graphs as a parabola, looking like this:
\n" ); document.write( " $\"graph%28200%2C200%2C-0.1%2C0.9%2C-0.1%2C0.9%2C4x-5x%5E2%29\"$ , with two zeros and a maximum exactly midway between them.
\n" ); document.write( "Re-writing it as $\"y=2x%281050-x%29\"$ shows you clearly that
\n" ); document.write( " $\"y=0\"$ for $\"x=0\"$ and $\"x=1050\"$ ,
\n" ); document.write( " $\"y%3E0\"$ in between, for $\"0%3Cx%3C1050\"$ ,
\n" ); document.write( "and the maximum is at $\"x=1050%2F2=525\"$
\n" ); document.write( "The dimensions that maximize the area are
\n" ); document.write( " $\"x=highlight%28525m%29\"$ for the length of each of the two fencing sides attached to the barn wall, and
\n" ); document.write( " $\"2100m-2x=2100m-2%28525m%29=2100m-1050m=highlight%281050m%29\"$ for the length of of the fencing side parallel to the barn wall.
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\n" ); document.write( "b) If we design something with four identical pens, like this:
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, where $\"x\"$ and $\"y\"$ are lengths in m,
\n" ); document.write( "we know that $\"x%2Ay=400m%5E2\"$ <---> $\"y=400%2Fx\"$ ,
\n" ); document.write( "and that makes the total length of fence needed, $\"y\"$ , in m
\n" ); document.write( " $\"y=5x%2B4%28400%2Fx%29=5x%2B1600%2Fx\"$ .
\n" ); document.write( "Maybe you are supposed to use a graphing calculator to find that the minimum for $\"y\"$ happens at approximately $\"x=17.888544\"$ ,
\n" ); document.write( "and you could tell the farmer to use 5 $\"17.9m\"$ length of fencing perpendicular to the barn wall,
\n" ); document.write( "attached to a $\"400m%5E2%2F%2217.9+m%22=approximately\"$ $\"22.35m\"$ length parallel to the wall.
\n" ); document.write( "
\n" ); document.write( "Using calculus, you would find that the derivative is
\n" ); document.write( " $\"dy%2Fdx=5-1600%2Fx%5E2=5-%2840%2Fx%29%5E2\"$ .
\n" ); document.write( "As $\"dy%2Fdx%3C0\"$ for $\"-40%2Fsqrt%285%29%3Cx%3C40%2Fsqrt%285%29\"$ ,
\n" ); document.write( " $\"dy%2Fdx=0\"$ for $\"x=40%2Fsqrt%285%29\"$ , and $\"dy%2Fdx%3E0\"$ for $\"x%3E40%2Fsqrt%285%29\"$ ,
\n" ); document.write( "the function decreases for $\"x%3E0\"$ to a minimum at $\"x=40%2Fsqrt%285%29=approximately\"$ $\"17.888544\"$ , and increases for $\"x%3E40%2Fsqrt%285%29\"$ .
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\n" ); document.write( "If there is another alternative approach, let me know. \n" ); document.write( "

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