document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1153924>Question 1153924</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The relevant manufacturing data are given in the table:\r<BR>\n" );
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document.write( "LABOR HRS<BR>\n" );
document.write( "                                                               MAX LABOR HRS AVAILABLE PER DAY<BR>\n" );
document.write( "DEPT	   Trick Ski  	Slalom Ski	<BR>\n" );
document.write( "Fabricating	8	            6                                  264<BR>\n" );
document.write( "Finishing	        1	            1	                                 40 \r<BR>\n" );
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document.write( "​(A) If the profit on a trick ski is ​$40 and the profit on a slalom ski is ​$50​, how many of each type of ski should be manufactured each day to realize a maximum​ profit? What is the maximum​ profit? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #776270 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=776270\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The other tutor provided a thorough solution to the problem using the standard process: graph the feasibility region and evaluate the profit function at each corner.<br><BR>\n" );
document.write( "In fact that is usually not necessary.<br><BR>\n" );
document.write( "The method described below can usually get you to the solution to this kind of problem faster and with less work.<br><BR>\n" );
document.write( "The given constraints give us the inequalities<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B6y+%3C=+264 ALIGN=MIDDLE  ALT=\"8x%2B6y+%3C=+264\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By+%3C=+40 ALIGN=MIDDLE  ALT=\"x%2By+%3C=+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And of course x and y have to be zero or positive.<br><BR>\n" );
document.write( "Consider the graphs of the boundary lines of those inequalities.  In slope-intercept form they are<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-4%2F3%29x%2B44 ALIGN=MIDDLE  ALT=\"y+=+%28-4%2F3%29x%2B44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%29x%2B40 ALIGN=MIDDLE  ALT=\"y+=+%28-1%29x%2B40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C50%2C-10%2C50%2C%28-4%2F3%29x%2B44%2C%28-1%29x%2B40%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C50%2C-10%2C50%2C%28-4%2F3%29x%2B44%2C%28-1%29x%2B40%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The corners of the feasibility region are (0,0), (0,40), (33,0) and the as yet undetermined point where the two constraint boundary line intersect.<br><BR>\n" );
document.write( "In the standard solution process, you would determine that 4th point and evaluate the profit function at each corner.  That is not necessary.<br><BR>\n" );
document.write( "Instead, consider the slope of the profit function.  The function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40x%2B50y+=+P ALIGN=MIDDLE  ALT=\"40x%2B50y+=+P\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In slope-intercept form, that is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-4%2F5%29x+%2B+P%2F50 ALIGN=MIDDLE  ALT=\"y+=+%28-4%2F5%29x+%2B+P%2F50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The maximum profit is going to occur when a line with slope -4/5 just touches a corner of the feasibility region, rather than passing through it.<br><BR>\n" );
document.write( "Comparing the slope of the profit function to the slopes of the two constraint boundary lines, it is clear that the corner of the feasibility region that will yield the maximum profit is (0,40).<br><BR>\n" );
document.write( "So the maximum profit is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%280%29%2B50%2840%29+=+2000 ALIGN=MIDDLE  ALT=\"40%280%29%2B50%2840%29+=+2000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
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