document.write( "Question 1132214: Marie is going to build a rectangular pen for her two dogs. She has 180 feet of fencing. To keep the dogs separate, she plans to put fencing down the middle of the pen to split the large rectangle into two smaller rectangles. What are the dimensions and area of the largest pen area she can use to accommodate both dogs? \n" ); document.write( "

Algebra.Com's Answer #749178 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "With a fence down the middle to keep the two dogs separate, the fencing must cover the length of the pen twice and the width three times:

\n" ); document.write( " $\"2L%2B3W+=+180\"$ --> $\"L+=+%28180-3W%29%2F2+=+90-%283%2F2%29W\"$

\n" ); document.write( "The total area of the pen is length times width:

\n" ); document.write( " $\"A+=+LW+=+W%2890-%283%2F2%29W%29+=+90W-%283%2F2%29W%5E2\"$

\n" ); document.write( "If you know calculus, find where the derivative of the area function is zero:

\n" ); document.write( " $\"dA%2FdW+=+90-3W\"$
\n" ); document.write( " $\"90-3W+=+0\"$
\n" ); document.write( " $\"W+=+30\"$

\n" ); document.write( "The maximum area is when W = 30, which makes L = 90-(3/2)(30) = 45.

\n" ); document.write( "If you don't know calculus, you can find the maximum area by finding the vertex of the parabolic area function.

\n" ); document.write( " $\"A+=+90W-%283%2F2%29W%5E2\"$
\n" ); document.write( " $\"W+=+-b%2F%282a%29+=+-90%2F-3+=+30\"$

\n" ); document.write( "Then, as before, W = 30 leads to L = 45.

\n" ); document.write( "So the dimensions for the greatest total area are 30 feet by 45 feet; the total area is 30*45 = 1350 square feet. \n" ); document.write( "

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