document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161281>Question 1161281</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A person needed a new rain spout, but instead of buying one, she <BR>\n" );
document.write( "decided to make her own by bending up an equal amount on both sides<BR>\n" );
document.write( " of a piece of sheet metal. If the sheet metal was 20 inches long, <BR>\n" );
document.write( "how much would she have to bend up on both sides so that the area <BR>\n" );
document.write( "would be 50 square inches? If you have two answers, type in the larger<BR>\n" );
document.write( " of your answers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #784922 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=784922\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What area? The bottom area? The cross section?<BR>\n" );
document.write( "Is the sheet of metal square? Rectangular in shape?<BR>\n" );
document.write( "A good design is like this, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28240%2C+290%2C-2%2C22%2C-2%2C27%2C%0D%0Arectangle%280%2C0%2C20%2C25%29%2C%0D%0Ared%28line%284%2C0%2C4%2C25%29%29%2Cred%28line%2816%2C0%2C16%2C25%29%29%2C%0D%0Aarrow%287%2C3%2C0%2C3%29%2Carrow%2813%2C3%2C20%2C3%29%2C%0D%0Alocate%287%2C4%2C%2220%28inch%29%22%29%2Clocate%288%2C15%2C20-2x%29%2C%0D%0Aarrow%2810%2C15%2C4%2C15%29%2Carrow%2810%2C15%2C16%2C15%29%2C%0D%0Ared%28arrow%280%2C12%2C4%2C12%29%29%2Cred%28arrow%284%2C12%2C0%2C12%29%29%2C%0D%0Ared%28arrow%2816%2C12%2C20%2C12%29%29%2Cred%28arrow%2820%2C12%2C16%2C12%29%29%2C%0D%0Alocate%281.5%2C12%2Cred%28x%29%29%2Clocate%2818.5%2C12%2Cred%28x%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >with the bends marked as red lines, at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches from the edges.<BR>\n" );
document.write( "Bending up the sides so they are perpendicular to the middle section,<BR>\n" );
document.write( "you would get a channel to carry the water.<BR>\n" );
document.write( "(Considering other bending angles could not be the intention of the problem, as it would complicate it and lead to an infinite number of solutions)<BR>\n" );
document.write( " <BR>\n" );
document.write( "The cross-section would look like this: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C80%2C2%2C18%2C-2%2C6%2C%0D%0Aline%284.2%2C0%2C15.8%2C0%29%2Cline%284%2C0.2%2C4%2C4%29%2Cline%2816%2C4%2C16%2C0.2%29%2C%0D%0Ared%28circle%284%2C0%2C0.1%29%29%2Cred%28circle%2816%2C0%2C0.1%29%29%2C%0D%0Ared%28circle%284%2C0.05%2C0.2%29%29%2Cred%28circle%2816%2C0.05%2C0.2%29%29%2C%0D%0Alocate%288%2C0%2C20-2x%29%2Clocate%2816.3%2C2.8%2Cred%28x%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of the rectangular cross-section is a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   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%29=x%2820-2x%29 ALIGN=MIDDLE  ALT=\"A%28x%29=x%2820-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=inch%5E2 ALIGN=MIDDLE  ALT=\"inch%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=-2x%5E2-20x ALIGN=MIDDLE  ALT=\"A%28x%29=-2x%5E2-20x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=inch%5E2 ALIGN=MIDDLE  ALT=\"inch%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The greater that area, the more water that rain spout can carry per minute.\r<BR>\n" );
document.write( "\n" );
document.write( "For a cross-section area of 50 square inches, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > should satisfy<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2-20x=50 ALIGN=MIDDLE  ALT=\"-2x%5E2-20x=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that is the quadratic equation to solve.<BR>\n" );
document.write( "(The hint about \"two answers\" suggested a quadratic equation).<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you have been shown a way to solve a problem like that in class,<BR>\n" );
document.write( "just do it as your teacher suggests.<BR>\n" );
document.write( "If that involves transforming that equation into a preferred form,<BR>\n" );
document.write( "and/or applying a memorized formula, pleasing the teacher is advisable.<BR>\n" );
document.write( "Solving it by factoring or by \"completing the square' may be also accepted, and it is easy enough in this case.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you are left to your own resources, you could try different values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > aiming to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=50 ALIGN=MIDDLE  ALT=\"A%28x%29=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those values must be between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because the lengths <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20-2x ALIGN=MIDDLE  ALT=\"20-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be positive).\r<BR>\n" );
document.write( "\n" );
document.write( "The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29 ALIGN=MIDDLE  ALT=\"A%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic function with a nice symmetrical graph called a parabola.<BR>\n" );
document.write( "The function has a maximum (represented by the top of that symmetrical graph.<BR>\n" );
document.write( "The value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=50 ALIGN=MIDDLE  ALT=\"A=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be reached at the maximum (one solution),<BR>\n" );
document.write( "or at two points to either side of that maximum (2 solutions),<BR>\n" );
document.write( "or never (if the maximum value is less than 50).<BR>\n" );
document.write( "You realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=2x%2810-x%29 ALIGN=MIDDLE  ALT=\"A%28x%29=2x%2810-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "so its value is zero at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10 ALIGN=MIDDLE  ALT=\"x=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(obvious if you thinks what that means for the cross-section drawing).<BR>\n" );
document.write( "Halfway between those two symmetrical points on the graph is the maximum,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%280%2B10%29%2F2-5 ALIGN=MIDDLE  ALT=\"x=%280%2B10%29%2F2-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=2%2A5%2810-5%29=2%2A5%2A5=50 ALIGN=MIDDLE  ALT=\"A%28x%29=2%2A5%2810-5%29=2%2A5%2A5=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the get a cross-section area of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50 ALIGN=MIDDLE  ALT=\"50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=inch%5E2 ALIGN=MIDDLE  ALT=\"inch%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you need to bend up <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%285inches%29 ALIGN=MIDDLE  ALT=\"highlight%285inches%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on both sides,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50 ALIGN=MIDDLE  ALT=\"50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=inch%5E2 ALIGN=MIDDLE  ALT=\"inch%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the maximum cross section area possible. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );