document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155269>Question 1155269</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer of wooden articles produces tables and chairs which require two types of inputs mainly, the being wood and labour. The manufacturer knows that for a table 3 units of wood and 1 unit of labour are required while for a chair they are 2 units each. The profit from each table is 20 USD while it is 16 USD for each chair. The total available resources for the manufacture are 150 units of wood and 75 units of labour. The minimum demand for chairs is 10 and the minimum demand for tables is 8.  The manufacturer wants to maximize his profit by distributing his resources for tables and chairs. Formulate the problem mathematically. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #777871 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=777871\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Note that the problem says \"Formulate the problem mathematically.\"<br><BR>\n" );
document.write( "It does not say anything about solving the problem....<br><BR>\n" );
document.write( "Assuming the problem IS supposed to be solved, here is a shortcut you can use for most problems like this where there are constraints on only two variables.<br><BR>\n" );
document.write( "(1) Plot the two major constraint boundary lines.<BR>\n" );
document.write( "(2) Determine the slope of each boundary line and the slope of the constraint line.<br><BR>\n" );
document.write( "The point of the feasibility region where the objective function is maximized is where a line with the slope of the objective function just touches the feasibility region.  That point can be determine by comparing the three slopes.<br><BR>\n" );
document.write( "For this problem, the major constraint boundary lines are<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2y+=+150 ALIGN=MIDDLE  ALT=\"3x%2B2y+=+150\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-3%2F2%29x%2B75 ALIGN=MIDDLE  ALT=\"y+=+%28-3%2F2%29x%2B75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y+=+75 ALIGN=MIDDLE  ALT=\"x%2B2y+=+75\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F2%29x%2B37.5 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F2%29x%2B37.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The objective function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+20x%2B16y ALIGN=MIDDLE  ALT=\"P+=+20x%2B16y\"   DISABLED_style= \"max-width:90%; height:auto;\" > which has a slope of -5/4.<br><BR>\n" );
document.write( "Because -5/4 is between -1/2 and -3/2, the point where the objective function is maximized is where the two major constraint lines intersect.<br><BR>\n" );
document.write( "So there is no need to evaluate the objective function at any point other than that point of intersection.<br><BR>\n" );
document.write( "Algebra shows the point of intersection to be (x,y) = (37.5,18.75).<br><BR>\n" );
document.write( "Note the non-integer values are probably okay if we consider the given data to be average daily production -- so that, for example, the maximum profit over 4 days is if 150 tables and 75 chairs are produced.<br><BR>\n" );
document.write( "Note also that we need to make sure the solution we found satisfies the other constraints that, up to this point, we have ignored.<br><BR>\n" );
document.write( "They do, so we have our answer.<br><BR>\n" );
document.write( "ANSWER: 37.5 tables and 18.75 chairs (per day?) for a maximum (daily) profit of 20(37.5)+16(18.75) = 750+300 = 1050 USD.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );