document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=615351>Question 615351</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A farmer decides to enclose a rectangular field using the side of a barn as one side of the rectangle. The figure below shows the fenced area he wants to make.\r<BR>\n" );
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document.write( "If he has 2800 feet of fencing, what values of x and y will maximize the enclosed area?\r<BR>\n" );
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document.write( "x = ? feet<BR>\n" );
document.write( "y = ? feet  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #387125 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=387125\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x = the lengths of the sides of the field that are perpendicular to the barn.<BR>\n" );
document.write( "Let y = the length of the side of the field that is parallel to the barn.<br><BR>\n" );
document.write( "With x and y we can now express the perimeter of the field:<BR>\n" );
document.write( "x + x + y = 2800<BR>\n" );
document.write( "which simplifies to<BR>\n" );
document.write( "2x + y = 2800<br><BR>\n" );
document.write( "The area of the field will be:<BR>\n" );
document.write( "A = x*y<br><BR>\n" );
document.write( "TO solve this we start by solving the perimeter equation for y. Subtracting 2x from each side we get:<BR>\n" );
document.write( "y = 2800 - 2x<br><BR>\n" );
document.write( "We can substitute this into the area equation for the y:<BR>\n" );
document.write( "A = x*(2800 - 2x)<BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+2800x+-+2x%5E2 ALIGN=MIDDLE  ALT=\"A+=+2800x+-+2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-2x%5E2+%2B+2800x ALIGN=MIDDLE  ALT=\"A+=+-2x%5E2+%2B+2800x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Assuming you are not in a Calculus class, the next step is to recognize that the Area equation is the equation of a parabola. And, since the coefficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is negative, this parabola opens downward. If we picture such a parabola, we should be able to realize that it will have a maximum (or highest) point. This will be the vertex of the parabola. So we want to find the x coordinate for the vertex of the parabola.<br><BR>\n" );
document.write( "You may have learned that the x coordinate of the vertex of a parabola is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since the b = 2800  and the a = -2, the x coordinate of our parabola will be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%282800%29%2F%282%28-2%29%29 ALIGN=MIDDLE  ALT=\"x+=+-%282800%29%2F%282%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%282800%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"x+=+-%282800%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 700<br><BR>\n" );
document.write( "So the value for x that gives us the greatest area is 700. We can use<BR>\n" );
document.write( "y = 2800 - 2x<BR>\n" );
document.write( "to find the y value:<BR>\n" );
document.write( "y = 2800 - 2(700)<BR>\n" );
document.write( "y = 2800 - 1400<BR>\n" );
document.write( "y = 1400<br><BR>\n" );
document.write( "So the maximum area can be achieved if the sides perpendicular to the barn are 700 feet and the side parallel to the barn is 1400 feet.<br><BR>\n" );
document.write( "P.S. This solution means the barn must be 1400 feet long itself in order for it to be the 4th side of the field. That's an awfully long barn!!<BR>\n" );
document.write( " </SPAN>\n" );
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