document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012279>Question 1012279</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 20 true/false questions and 20 multiple choice questions on a test.\r<BR>\n" );
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document.write( "A correct answer to a true false question earns 10 points.<BR>\n" );
document.write( "A correct answer to a multiple choice question earns 12 points.\r<BR>\n" );
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document.write( "The test makers determined that it takes, on average, 3 minutes to answer a true/false question and 4 minutes to answer a multiple choice question. Students have 1 hour to answer to answer at most 18 questions of their choice. How many of each kind of question should a student answer correctly to get the greatest possible score? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #628249 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=628249\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= number of true/false questions answered<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= number of multiple choice questions answered<BR>\n" );
document.write( "Since \"A correct answer to a true false question earns 10 points\" and<BR>\n" );
document.write( "\"A correct answer to a multiple choice question earns 12 points\",<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=score=10x%2B12y ALIGN=MIDDLE  ALT=\"score=10x%2B12y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is a linear function on two variables.<BR>\n" );
document.write( "We have some constraints that limit the domain of that function.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are implicit constraints.\r<BR>\n" );
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document.write( "\"Students have ... to answer to answer at most 18 questions of their choice\",<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By%3C=18 ALIGN=MIDDLE  ALT=\"x%2By%3C=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <BR>\n" );
document.write( "is an explicit constraint that sounds strange to a rat-race accustomed American,<BR>\n" );
document.write( "who would expect that instead students would be required to rush to answer as many questions as possible,<BR>\n" );
document.write( "and maybe to answer a minimum number of questions, as in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By%3E=18 ALIGN=MIDDLE  ALT=\"x%2By%3E=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Maybe the students are only given credit for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" > questions in order to encourage them to take time and work carefully, instead of rushing to answer as many questions as possible.<BR>\n" );
document.write( "The expected pace, \"it takes, on average, 3 minutes to answer a true/false question and 4 minutes to answer a multiple choice question\", means that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y ALIGN=MIDDLE  ALT=\"3x%2B4y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= time (in minutes) needed to answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > true/false questions and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > multiple choice questions.<BR>\n" );
document.write( "That leads to another explicit constraint of the problem, because the student can use at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes (1 hour).<BR>\n" );
document.write( "That constraint can be written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=60 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=60 ALIGN=MIDDLE  ALT=\"3x%2B4y=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > obviously passes through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=0%2Cy=18%29 ALIGN=MIDDLE  ALT=\"system%28x=0%2Cy=18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=0%2Cx=20%29 ALIGN=MIDDLE  ALT=\"system%28y=0%2Cx=20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so we graph it as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C250%2C-3%2C27%2C-3%2C24%2Cgrid%281%29%2C%0D%0Aline%28-4%2C18%2C24%2C-3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=0%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28x=0%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=60 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the graph for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%3C=60 ALIGN=MIDDLE  ALT=\"3x%2B4y%3C=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C250%2C-3%2C27%2C-3%2C24%2Cgrid%281%29%2C%0D%0Agraph%28300%2C250%2C-3%2C27%2C-3%2C24%2C3x%2B4y%3C=60%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , includes the origin, the point (0,0) . <BR>\n" );
document.write( "Working in the same manner, we find that the constraint <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By%3C=18 ALIGN=MIDDLE  ALT=\"x%2By%3C=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be graphed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C250%2C-3%2C27%2C-3%2C24%2Cgrid%281%29%2C%0D%0Agraph%28300%2C250%2C-3%2C27%2C-3%2C24%2Cx%2By%3C=18%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By%3E=18 ALIGN=MIDDLE  ALT=\"x%2By%3E=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be graphed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C250%2C-3%2C27%2C-3%2C24%2Cgrid%281%29%2C%0D%0Agraph%28300%2C250%2C-3%2C27%2C-3%2C24%2Cx%2By%3E=18%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "With the constraints <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%3E=0%2Cy%3E=0%2C3x%2B4y%3C=60%2Cx%2By%3C=18%29 ALIGN=MIDDLE  ALT=\"system%28x%3E=0%2Cy%3E=0%2C3x%2B4y%3C=60%2Cx%2By%3C=18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we have four inequalities that determine a feasibility space that is a convex polygon. It is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C250%2C-3%2C27%2C-3%2C24%2Cgrid%281%29%2C%0D%0Agreen%28line%280%2C15%2C12%2C6%29%29%2Cgreen%28line%2812%2C6%2C18%2C0%29%29%2C%0D%0Agreen%28line%280%2C0%2C0%2C15%29%29%2Cgreen%28line%280%2C0%2C18%2C0%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , with vertices at (0,0) , (0,15) , (12,6) , and (18,0) .<BR>\n" );
document.write( " <BR>\n" );
document.write( "An extreme value (maximum or minimum) of linear function, such as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=score=10x%2B12y ALIGN=MIDDLE  ALT=\"score=10x%2B12y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , defined on a domain that is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=score=10x%2B12y ALIGN=MIDDLE  ALT=\"score=10x%2B12y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "occurs at a vertex of the domain, or all along an edge between two vertices,<BR>\n" );
document.write( "so we only need to check the scores for the vertices.<BR>\n" );
document.write( "At (0,0) , obviously <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=score=0 ALIGN=MIDDLE  ALT=\"score=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At (0,15) , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=0%2Cy=15%29 ALIGN=MIDDLE  ALT=\"system%28x=0%2Cy=15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=score=10%2A0%2B12%2A15=180 ALIGN=MIDDLE  ALT=\"score=10%2A0%2B12%2A15=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At (18,0) , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=18%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28x=18%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=score=10%2A18%2B12%2A0=180 ALIGN=MIDDLE  ALT=\"score=10%2A18%2B12%2A0=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At (12,6) , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28x=12%2Cy=6%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28x=12%2Cy=6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=score=10%2A12%2B12%2A6=120%2B72=192 ALIGN=MIDDLE  ALT=\"score=10%2A12%2B12%2A6=120%2B72=192\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the student gets the greatest possible score. </SPAN>\n" );
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