document.write( "Question 167388: Find the dimensions of the rectangular field of maximum area which can be enclosed with 400 feet of fence. \n" ); document.write( "
Algebra.Com's Answer #123284 by nerdybill(7384)\"\" \"About 
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Find the dimensions of the rectangular field of maximum area which can be enclosed with 400 feet of fence.
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\n" ); document.write( "Let w = width of field
\n" ); document.write( "then
\n" ); document.write( "(400-2w)/2 = length of field
\n" ); document.write( "(200-w) = length of field
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\n" ); document.write( "area = w(200-w)
\n" ); document.write( "area = 200w-w^2
\n" ); document.write( "area = -w^2+200w
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\n" ); document.write( "Looking at the coefficient for the w^2 term, we see that it is negative. This indicates that the parabola opens downward and finding the vertex will give you the \"maximum\".
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\n" ); document.write( "Standard vertex form is:
\n" ); document.write( "y= a(x-h)^2+k
\n" ); document.write( "where
\n" ); document.write( "(h,k) is the vertex
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\n" ); document.write( "Convert our equation into that form by \"completing the square\":
\n" ); document.write( "area = -w^2+200w
\n" ); document.write( "area = -(w^2-200w)
\n" ); document.write( "area = -(w^2-200w+10000) + 10000
\n" ); document.write( "area = -(w-100)^2 + 10000
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\n" ); document.write( "From the above, we see that the vertex is:
\n" ); document.write( "(h,k) = (100, 10000)
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\n" ); document.write( "This says that when the width=100 feet, the area will be maximized at 10000 square feet.
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\n" ); document.write( "Solution:
\n" ); document.write( "width = 100 feet
\n" ); document.write( "length = 200-w = 200-100 = 100 feet\r
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