document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163322>Question 1163322</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A two-pen corral is to be built. The outline of the corral forms two<BR>\n" );
document.write( "identical adjoining rectangles. If there is 120 m of fencing available and<BR>\n" );
document.write( "the fence width cannot be less than 6 m, what dimensions of the corral will<BR>\n" );
document.write( "maximize<BR>\n" );
document.write( "the enclosed area? [round all answers to one decimal places)<BR>\n" );
document.write( "  \r<BR>\n" );
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document.write( "In the rectangular corral Horizontal line is the length and vertical line is the width.\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #787392 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=787392\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let the width be x.<br><BR>\n" );
document.write( "Since the corral is divided in half, there are three fences of length x; that's a total fence length of 3x.<br><BR>\n" );
document.write( "The remaining length of fence, 120-3x, is the two other sides of the corral, so the length of the corral is (120-3x)/2, or 60-1.5x.<br><BR>\n" );
document.write( "The area of the corral is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2860-1.5x%29+=+-1.5x%5E2%2B60x ALIGN=MIDDLE  ALT=\"x%2860-1.5x%29+=+-1.5x%5E2%2B60x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a quadratic expression of the form ax^2+bx+c; its value is maximized when x is equal to -b/2a.<br><BR>\n" );
document.write( "In this quadratic expression, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29+=+-60%2F-3+=+20 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29+=+-60%2F-3+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the width that maximizes the volume is x=20 (which satisfies the condition that it must be at least 6).  And then the length is (120-3x)/2 = 30.<br><BR>\n" );
document.write( "ANSWER: length 30 and width 20 maximizes the area.<br><BR>\n" );
document.write( "Note that the total lengths of fencing in the two directions are equal.  This is always the case, no matter how many adjacent corrals the whole corral is divided into.<br><BR>\n" );
document.write( "For example, if you have 600 feet of fencing and you are dividing the long corral into 5 adjacent sections (using 6 sections of fence width-wise), then the maximum area is with 300 feet of fencing in each direction -- which means a length of 300/2 = 150 feet and a width of 300/6 = 50 feet.<br><BR>\n" );
document.write( "So to an experienced problem solver, the solution to your problem would go like this:<br><BR>\n" );
document.write( "Total fence length: 120 feet<BR>\n" );
document.write( "Length of fencing in each direction: 120/2 = 60 feet<BR>\n" );
document.write( "Length of corral for maximum area: 60/2 = 30 feet<BR>\n" );
document.write( "Width of corral for maximum area: 60/3 = 20 feet<br><BR>\n" );
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