document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=56158>Question 56158</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Amanda has 400 feet of lumber to frame a rectangular patio.  She wants to maximize the area of her patio.  What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation.  Use the vertex form to find the maximum area. \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #38091 by </B><A HREF=/tutors/aboutme.mpl?userid=funmath><B>funmath(2933)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=funmath><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=38091\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Amanda has 400 feet of lumber to frame a rectangular patio.  She wants to maximize the area of her patio.  What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation.  Use the vertex form to find the maximum area. \r<BR>\n" );
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document.write( "Answer: A=10,000 square ft\r<BR>\n" );
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document.write( "Let length=x<BR>\n" );
document.write( "Let width=y<BR>\n" );
document.write( "Perimeter=400<BR>\n" );
document.write( "2(length)+2(width)=Perimeter<BR>\n" );
document.write( "2x+2y=400<BR>\n" );
document.write( "2y=400-2x<BR>\n" );
document.write( "2y/2=400/2-2x/2<BR>\n" );
document.write( "y=200-x<BR>\n" );
document.write( "A=length*width<BR>\n" );
document.write( "A=xy  but y=200-x so<BR>\n" );
document.write( "A=x(200-x)<BR>\n" );
document.write( "Vertex form is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"A=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where (h,k) is the vertex.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=200x-x%5E2 ALIGN=MIDDLE  ALT=\"A=200x-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-x%5E2%2B200x ALIGN=MIDDLE  ALT=\"A=-x%5E2%2B200x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-%28x%5E2-200x%29 ALIGN=MIDDLE  ALT=\"A=-%28x%5E2-200x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  We need to create a perfect square.  We do that by adding (-200/2)^2=(100)^2=10000 to the inside of the parenthesis.  Because of the - infront of the parenthesis we are really taking 10000 away, so we have to add the same thing on the outside of the parenthesis so that we are in reality only adding 0. <BR>\n" );
document.write( "A=-(x^2-200x+____)+____<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-%28x%5E2-200x%2B10000%29%2B10000 ALIGN=MIDDLE  ALT=\"A=-%28x%5E2-200x%2B10000%29%2B10000\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Now we have to factor the perfect square we created. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a%5E2-2ab%2Bb%5E2=%28a-b%29%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28a%5E2-2ab%2Bb%5E2=%28a-b%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  b^2=10000, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=sqrt%2810000%29=100 ALIGN=MIDDLE  ALT=\"b=sqrt%2810000%29=100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-%28x-100%29%5E2%2B10000 ALIGN=MIDDLE  ALT=\"A=-%28x-100%29%5E2%2B10000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The vertex is (100, 10000), that means that the maximum x (length) is 100 ft, and the maximum A (area) is 10000 ft^2<BR>\n" );
document.write( "I hope this is a little more clear.<BR>\n" );
document.write( "Happy Calculating!!! </SPAN>\n" );
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