document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=636765>Question 636765</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help. Due tomorrow.<BR>\n" );
document.write( "1.)A box is to be formed by cutting squares measuring 2 cm on a side from a square piece of cardboard and then folding the sided. If the volume of the box is to be 392 cm squared, what is the original size of the cardboard?<BR>\n" );
document.write( "2.)Manuel is constructing a model house. He wants each window to have an area of 315cm squared, and he wants the length of each window to be 6 cm more than the width. What are the dimensions of each window?<BR>\n" );
document.write( "3.)A circle has a radius of 20 cm. By how much must the radius be increased so that the area will increase to 144 pie cm squared?<BR>\n" );
document.write( "4.)An airplane flies 900 miles against a headwind if 25 miles per hour. The plane took 15 minutes longer for this flight than with a tailwind of 25 miles per hour. How fast could the plane fly in still air? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #401256 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=401256\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.)The box is formed by cutting squares measuring 2 cm on a side from a square piece of cardboard That looks like this, with the cutting lines marked in red:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28360%2C300%2C-10%2C12%2C-10%2C10%2C%0D%0Ared%28rectangle%287%2C7%2C9%2C9%29%29%2C%0D%0Ared%28rectangle%28-7%2C7%2C-9%2C9%29%29%2C%0D%0Ared%28rectangle%287%2C-7%2C9%2C-9%29%29%2C%0D%0Ared%28rectangle%28-7%2C-7%2C-9%2C-9%29%29%2C%0D%0Arectangle%28-9%2C-9%2C9%2C9%29%2C%0D%0Ablue%28rectangle%28-7%2C-7%2C7%2C7%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The folding lines, shown in blue, show the central square that will be the base of the box.<BR>\n" );
document.write( "The box sides (the folded flaps on all four sides) will 2 cm high, and that will be the height of the box.<BR>\n" );
document.write( "The volume of the box, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" >, is the surface area of the base, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >, times the height, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the box.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=A%2Ah ALIGN=MIDDLE  ALT=\"V=A%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=V%2Fh ALIGN=MIDDLE  ALT=\"A=V%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the volume of the box is to be 392 cubic centimeters, then the surface area  of the base of the box is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >=(392 cubic centimeters)/(2 centimeters)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=392cm%5E3%2F2cm=196cm%5E2 ALIGN=MIDDLE  ALT=\"392cm%5E3%2F2cm=196cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=196 square centimeters. <BR>\n" );
document.write( "What is the original size of the cardboard?<BR>\n" );
document.write( "THE FIFTH GRADER SOLUTION:<BR>\n" );
document.write( "The area of the square that is the base of the box is side of the base of the box squared.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=196=14%5E2 ALIGN=MIDDLE  ALT=\"196=14%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the base of the box is a 14 cm by 14 cm square.<BR>\n" );
document.write( "With 2 cm of cardboard to each side (4 extra cm), the width of the square piece of cardboard, before cutting was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14cm%2B4cm=highlight%2818cm%29 ALIGN=MIDDLE  ALT=\"14cm%2B4cm=highlight%2818cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "IF YOU KNOW ENOUGH ALGEBRA TO COMPLICATE THE PROBLEM:<BR>\n" );
document.write( "You start by saying, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm be the length of each side of the square piece of cardboard, which looks like this, with the cutting lines marked in red:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28360%2C300%2C-10%2C12%2C-10%2C10%2C%0D%0Ared%28rectangle%287%2C7%2C9%2C9%29%29%2C%0D%0Ared%28rectangle%28-7%2C7%2C-9%2C9%29%29%2C%0D%0Ared%28rectangle%287%2C-7%2C9%2C-9%29%29%2C%0D%0Ared%28rectangle%28-7%2C-7%2C-9%2C-9%29%29%2C%0D%0Arectangle%28-9%2C-9%2C9%2C9%29%2C%0D%0Ablue%28rectangle%28-7%2C-7%2C7%2C7%29%29%2C%0D%0Alocate%287.8%2C7%2C2%29%2C+locate%287.1%2C8.5%2C2%29%2C%0D%0Alocate%28-8.3%2C7%2C2%29%2C+locate%28-7.5%2C8.5%2C2%29%2C%0D%0Alocate%289.8%2C0.5%2Cx%29%2Clocate%285%2C0.5%2Cx-4%29%2C%0D%0Aarrow%285.5%2C0.5%2C5.5%2C7%29%2Carrow%285.5%2C-0.5%2C5.5%2C-7%29%2C%0D%0Aarrow%2810%2C0.5%2C10%2C9%29%2Carrow%2810%2C-0.5%2C10%2C-9%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The folding lines, shown in blue, show the central square that will be the base of the box.<BR>\n" );
document.write( "That length of each side of that square base of the box is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2-2=x-4 ALIGN=MIDDLE  ALT=\"x-2-2=x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm<BR>\n" );
document.write( "That surface area of the base of the box (in square cm) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28x-4%29%5E2=x%5E2-8x%2B16 ALIGN=MIDDLE  ALT=\"A=%28x-4%29%5E2=x%5E2-8x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "The height of the box is 2 cm.<BR>\n" );
document.write( "The volume of the box (area of the base times height) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-8x%2B16%29%2A2=392 ALIGN=MIDDLE  ALT=\"%28x%5E2-8x%2B16%29%2A2=392\"   DISABLED_style= \"max-width:90%; height:auto;\" > square centimeters.<BR>\n" );
document.write( "If we are smart, we divide both sides by 2 now to get a simpler equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-8x%2B16%29%2A2%2F2=392%2F2 ALIGN=MIDDLE  ALT=\"%28x%5E2-8x%2B16%29%2A2%2F2=392%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16=196 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16=196\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we subtract 196 from both sides, to get a quadratic equation in standard form.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16-196=196-196 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16-196=196-196\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x-180=0 ALIGN=MIDDLE  ALT=\"x%5E2-8x-180=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To solve that quadratic equation, if we are not good at factoring, we have to \"complete the square\", or use the dreaded quadratic formula.<BR>\n" );
document.write( "If we are good at factoring, we realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x-180=%28x-18%29%28x%2B10%29 ALIGN=MIDDLE  ALT=\"x%5E2-8x-180=%28x-18%29%28x%2B10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and find that the solutions are <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=18 ALIGN=MIDDLE  ALT=\"x=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Since the length of the side of the original square piece of cardboard cannot be a negative number, the only solution is<BR>\n" );
document.write( "The length of the side of the original square piece of cardboard is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=18%29 ALIGN=MIDDLE  ALT=\"highlight%28x=18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm.<BR>\n" );
document.write( " <BR>\n" );
document.write( "2.)<BR>\n" );
document.write( "THE FIFTH GRADER SOLUTION:<BR>\n" );
document.write( "The fifth grader may factor to find that the prime factorization of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=315 ALIGN=MIDDLE  ALT=\"315\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=315=3%2A3%2A5%2A7 ALIGN=MIDDLE  ALT=\"315=3%2A3%2A5%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and would rearrange that to find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=315=%283%2A5%29%2A%283%2A7%29=15%2A21 ALIGN=MIDDLE  ALT=\"315=%283%2A5%29%2A%283%2A7%29=15%2A21\"   DISABLED_style= \"max-width:90%; height:auto;\" > to figure out that the windows are 15 cm by 21 cm.<BR>\n" );
document.write( "That fifth grader is smart and lucky, but we should try to use algebra, because the fifth grader approach would not work for a similar problem where the answers were not integers. (However, problems are often rigged so they have integer answers, and it is good to have a smart fifth grader around, so he/she can check your algebra answers).<BR>\n" );
document.write( "THE ALGEBRA SOLUTION:<BR>\n" );
document.write( "Let the width of each window be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm.<BR>\n" );
document.write( "Then the length of each window (in cm) would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B6 ALIGN=MIDDLE  ALT=\"x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The area (in square centimeters) of each window, calculated multiplying length times width, would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B6%29x=315 ALIGN=MIDDLE  ALT=\"%28x%2B6%29x=315\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We work with that equation to transform it into a quadratic equation in a form that we like better.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B6%29x=315 ALIGN=MIDDLE  ALT=\"%28x%2B6%29x=315\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x=315 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x=315\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x-315=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x-315=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can solve it by \"completing the square\", or by factoring, of by using the quadratic formula.<BR>\n" );
document.write( "Completing the square:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x=315 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x=315\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%2B9=315%2B9 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%2B9=315%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2=324 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2=324\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2=%28sqrt%28324%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2=%28sqrt%28324%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2=18%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2=18%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The two solutions would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3=-18 ALIGN=MIDDLE  ALT=\"x%2B3=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-21 ALIGN=MIDDLE  ALT=\"x=-21\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which does not make sense, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3=18 ALIGN=MIDDLE  ALT=\"x%2B3=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3-3=18-3 ALIGN=MIDDLE  ALT=\"x%2B3-3=18-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=15%29 ALIGN=MIDDLE  ALT=\"highlight%28x=15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the width of the windows is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815%29 ALIGN=MIDDLE  ALT=\"highlight%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm and the length is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2B6=highlight%2821%29 ALIGN=MIDDLE  ALT=\"15%2B6=highlight%2821%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm.<BR>\n" );
document.write( "Factoring:<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x-315=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x-315=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we realize that<BR>\n" );
document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=315=15%2A25 ALIGN=MIDDLE  ALT=\"315=15%2A25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21-15=6 ALIGN=MIDDLE  ALT=\"21-15=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x-315=%28x%2B21%29%28x-15%29 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x-315=%28x%2B21%29%28x-15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so we re-write the equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B21%29%28x-15%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B21%29%28x-15%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "One of those factor must be zero, and realize that the solution coming from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-15=0 ALIGN=MIDDLE  ALT=\"x-15=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=15%29 ALIGN=MIDDLE  ALT=\"highlight%28x=15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us a valid window width of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815%29 ALIGN=MIDDLE  ALT=\"highlight%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm, for a length of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2B6=highlight%2821%29 ALIGN=MIDDLE  ALT=\"15%2B6=highlight%2821%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm.<BR>\n" );
document.write( "On the other hand, making the other factor zero, leads to an invalid solution:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B21=0 ALIGN=MIDDLE  ALT=\"x%2B21=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-21 ALIGN=MIDDLE  ALT=\"x=-21\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because the width of the windows cannot be a negative number.<BR>\n" );
document.write( " <BR>\n" );
document.write( "3.)The area, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > of a circle of radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A=pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area (in square centimeters) of circle with a radius of 20 cm is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=pi%2A20%5E2=400pi ALIGN=MIDDLE  ALT=\"a=pi%2A20%5E2=400pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is more than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144pi ALIGN=MIDDLE  ALT=\"144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so either<BR>\n" );
document.write( "there is a typo somewhere in the problem, or<BR>\n" );
document.write( "it is a trick question, where they call a decrease in area from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400pi ALIGN=MIDDLE  ALT=\"400pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > down to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144pi ALIGN=MIDDLE  ALT=\"144pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is called<BR>\n" );
document.write( "an increase of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144%7D%7D+%7B%7B%7Bcm%5E2 ALIGN=MIDDLE  ALT=\"144%7D%7D+%7B%7B%7Bcm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400pi ALIGN=MIDDLE  ALT=\"400pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-256pi ALIGN=MIDDLE  ALT=\"-256pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "If the area of a circle with radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ar%5E2=144picm%5E2 ALIGN=MIDDLE  ALT=\"pi%2Ar%5E2=144picm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , dividing both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ar%5E2%2Fpi=144picm%5E2%2Fpi ALIGN=MIDDLE  ALT=\"pi%2Ar%5E2%2Fpi=144picm%5E2%2Fpi\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=144cm%5E2 ALIGN=MIDDLE  ALT=\"r%5E2=144cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=sqrt%28144cm%5E2%29=12cm ALIGN=MIDDLE  ALT=\"r=sqrt%28144cm%5E2%29=12cm\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12cm-20cm=highlight%28-8cm%29 ALIGN=MIDDLE  ALT=\"12cm-20cm=highlight%28-8cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I would say that the radius must be increased <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28-8cm%29 ALIGN=MIDDLE  ALT=\"highlight%28-8cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that the area will increase (by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-256pi ALIGN=MIDDLE  ALT=\"-256pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) to 144 pie cm squared?<BR>\n" );
document.write( " <BR>\n" );
document.write( "4.)If the speed of an airplane relative to the air is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour,<BR>\n" );
document.write( "the plane will move 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;\" > miles per hour with respect to the ground in still air.<BR>\n" );
document.write( "With a tail wind of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour, the same airplane will move at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B25 ALIGN=MIDDLE  ALT=\"x%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour with respect to the ground.<BR>\n" );
document.write( "Against a wind of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour, the same airplane will move at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-25 ALIGN=MIDDLE  ALT=\"x-25\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour with respect to the ground. An airplane flies 900 miles against a headwind of 25 miles per hour.<BR>\n" );
document.write( "We know that speed, time and distance are related like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=speed=distance%2Ftime ALIGN=MIDDLE  ALT=\"speed=distance%2Ftime\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=speed%2Atime=distance ALIGN=MIDDLE  ALT=\"speed%2Atime=distance\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=time=distance%2Fspeed ALIGN=MIDDLE  ALT=\"time=distance%2Fspeed\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we want to figure how long a trip will take, we divide distance by speed.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15+minute=15minute%281hour%2F60minute%29=0.25hour ALIGN=MIDDLE  ALT=\"15+minute=15minute%281hour%2F60minute%29=0.25hour\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the plane took 15 minutes longer to fly the same 900-mile distance against that wind than with the wind, the equation relating the times, in hours is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900%2F%28x-25%29=900%2F%28x%2B25%29%2B0.25 ALIGN=MIDDLE  ALT=\"900%2F%28x-25%29=900%2F%28x%2B25%29%2B0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To get rid of denominators, we multiply both sides of the equation time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-25%29%28x%2B25%29 ALIGN=MIDDLE  ALT=\"%28x-25%29%28x%2B25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28900%2F%28x-25%29%29%28x-25%29%28x%2B25%29=%28900%2F%28x%2B25%29%2B0.25%29%28x-25%29%28x%2B25%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900%28x%2B25%29=%28900%2F%28x%2B25%29%29%28x-25%29%28x%2B25%29%2B0.25%28x-25%29%28x%2B25%29 ALIGN=MIDDLE  ALT=\"900%28x%2B25%29=%28900%2F%28x%2B25%29%29%28x-25%29%28x%2B25%29%2B0.25%28x-25%29%28x%2B25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900%28x%2B25%29=900%28x-25%29%2B0.25%28x-25%29%28x%2B25%29 ALIGN=MIDDLE  ALT=\"900%28x%2B25%29=900%28x-25%29%2B0.25%28x-25%29%28x%2B25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900%28x%2B25%29=900%28x-25%29%2B0.25%28x%5E2-625%29 ALIGN=MIDDLE  ALT=\"900%28x%2B25%29=900%28x-25%29%2B0.25%28x%5E2-625%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900x%2B22500=900x-22500%2B0.25x%5E2-156.25%29 ALIGN=MIDDLE  ALT=\"900x%2B22500=900x-22500%2B0.25x%5E2-156.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-900x-22500 ALIGN=MIDDLE  ALT=\"-900x-22500\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides, or subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900x%2B22500 ALIGN=MIDDLE  ALT=\"900x%2B22500\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900x%2B22500-900x-22500=900x-22500%2B0.25x%5E2-156.25-900x-22500 ALIGN=MIDDLE  ALT=\"900x%2B22500-900x-22500=900x-22500%2B0.25x%5E2-156.25-900x-22500\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=-22500%2B0.25x%5E2-156.25-22500 ALIGN=MIDDLE  ALT=\"0=-22500%2B0.25x%5E2-156.25-22500\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.25x%5E2-45156.25=0 ALIGN=MIDDLE  ALT=\"0.25x%5E2-45156.25=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%280.25x%5E2-45156.25%29=4%2A0 ALIGN=MIDDLE  ALT=\"4%280.25x%5E2-45156.25%29=4%2A0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-180625=0 ALIGN=MIDDLE  ALT=\"x%5E2-180625=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28180625%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28180625%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=425%29 ALIGN=MIDDLE  ALT=\"highlight%28x=425%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles per hour. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );