document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=438013>Question 438013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Carl wants to construct three rectangular dog-training arenas side-by-side using a total of 340 feet of fencing. What should the overall length and width be in order to maximize the area of the three combined arenas? (let x represent the width).\r<BR>\n" );
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document.write( "I used the formula A=3xy to find the area and I came up with:<BR>\n" );
document.write( "A=3x(340-4x)/6\r<BR>\n" );
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document.write( "I also used the formula P=4x+6y to find the perimeter and I came up with:<BR>\n" );
document.write( "Y=(340-4x)/6\r<BR>\n" );
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document.write( "I don't know where to go from here. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #302991 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=302991\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "You are over-thinking the setup here.\r<BR>\n" );
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document.write( "Imagine 3 rectangles stacked on top of one another creating another, larger rectangle.\r<BR>\n" );
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document.write( "Let the width of this larger rectangle be represented by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  Let the length of the larger rectangle (the sum of the widths of the three small ones) be represented by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">\r<BR>\n" );
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document.write( "Now the perimeter of this larger rectangle is just <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\ +\ 2y\"> like any other rectangle, but remember that this is three rectangular areas so there are two more pieces of fence each of which measures <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">, so we can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2y\ +\ 4x\ =\ 340\">\r<BR>\n" );
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document.write( "to show the relationship between the length and width of the large rectangle and the amount of available fencing.\r<BR>\n" );
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document.write( "Next, the area of the large rectangle, which is the area we are asked to maximize is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x,y)\ =\ xy\">\r<BR>\n" );
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document.write( "But if we solve the fencing relationship for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> we can create a function for the area in terms of the width:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 170\ -\ 2x\">\r<BR>\n" );
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document.write( "Then, by substitution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ -2x^2\ +\ 170x\">\r<BR>\n" );
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document.write( "Which is a quadratic function of the form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ ax^2\ +\ bx\ + c\">\r<BR>\n" );
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document.write( "where\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ -2\ \ \">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\ =\ 170\ \ \">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c\ =\ 0\">\r<BR>\n" );
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document.write( "Since this is a quadratic with a negative lead coefficient, the graph is a parabola that opens downward, hence the vertex represents the maximum value of the function.\r<BR>\n" );
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document.write( "Use the formula for the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex of a parabola to find the overall width that maximizes the area:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_v = \frac{-b}{2a}\ =\ \frac{-170}{2(-2)}\ =\ 42.5\ \ \">feet\r<BR>\n" );
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document.write( "Substitute this value into the equation for the fencing:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2y\ +\ 4(42.5)\ =\ 340\">\r<BR>\n" );
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document.write( "and solve for the overall length:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 85\ \ \">feet\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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