document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198959>Question 1198959</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A factory makes two types of beds, type A and type B. Each month, a number a number of type A and a number of type B are produced. The following constraints control monthly production:<BR>\n" );
document.write( "No more than 50 beds of Type A and no more than 40 beds of type B can be made.<BR>\n" );
document.write( "At least 60 beds in all must be made.<BR>\n" );
document.write( "The maximum number of beds that can be produced is 80.\r<BR>\n" );
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document.write( "The profit on type A is Php300 and on type B is Php150. How many beds on both types must be produced to maximize the profit? What is the minimum profit? </SPAN>\n" );
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document.write( "Given Facts:<ol><li>No more than 50 beds of Type A can be made.</li><li>No more than 40 beds of type B can be made.</li><li>At least 60 beds in all must be made.</li><li>The maximum number of beds that can be produced is 80.</li><li>The profit on type A is Php300.</li><li>The profit on type B is Php150. </li></ol>x = number of beds of type A<BR>\n" );
document.write( "y = number of beds of type B<BR>\n" );
document.write( "where x,y are nonnegative integers\r<BR>\n" );
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document.write( "Fact 1 leads to the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C=+50 ALIGN=MIDDLE  ALT=\"x+%3C=+50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since x is nonnegative, we can further clarify that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C=+50 ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C=+50\"   DISABLED_style= \"max-width:90%; height:auto;\" > (i.e. x is between 0 and 50 inclusive of each endpoint)<BR>\n" );
document.write( "x is in the set {0,1,2,...,49,50}\r<BR>\n" );
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document.write( "Fact 2 leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3C=+40 ALIGN=MIDDLE  ALT=\"y+%3C=+40\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we can clarify to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+y+%3C=+40 ALIGN=MIDDLE  ALT=\"0+%3C=+y+%3C=+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y is in the set {0,1,2,...,39,40}\r<BR>\n" );
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document.write( "Fact 3 means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By+%3E=+60 ALIGN=MIDDLE  ALT=\"x%2By+%3E=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" > because \"at least 60\" means \"60 or more\".\r<BR>\n" );
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document.write( "Then fact 4 says <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By+%3C=+80 ALIGN=MIDDLE  ALT=\"x%2By+%3C=+80\"   DISABLED_style= \"max-width:90%; height:auto;\" > to put a ceiling on the production amount. <BR>\n" );
document.write( "The most beds that can be made is 80.\r<BR>\n" );
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document.write( "Fact 5 tells us 300x is the profit for just the type A beds.<BR>\n" );
document.write( "Fact 6 tells us 150y is the profit for just the type B beds.<BR>\n" );
document.write( "Combine those facts to get 300x+150y as the total profit for both beds.\r<BR>\n" );
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document.write( "The goal is to max out P = 300x+150y\r<BR>\n" );
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document.write( "System of inequalities<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%280+%3C=+x+%3C=+50%2C+0+%3C=+y+%3C=+40%2Cx%2By+%3E=+60%2Cx%2By%3C=80%29 ALIGN=MIDDLE  ALT=\"system%280+%3C=+x+%3C=+50%2C+0+%3C=+y+%3C=+40%2Cx%2By+%3E=+60%2Cx%2By%3C=80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Graph<BR>\n" );
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document.write( "The blue trapezoidal region represents the set of (x,y) points that satisfy all of the inequalities mentioned in the system above.<BR>\n" );
document.write( "Points on the boundary are part of the shaded solution set.\r<BR>\n" );
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document.write( "The corner points are<BR>\n" );
document.write( "A = (20, 40)<BR>\n" );
document.write( "B = (40, 40)<BR>\n" );
document.write( "C = (50, 30)<BR>\n" );
document.write( "D = (50, 10)<BR>\n" );
document.write( "Each corner point can be determined using algebra.<BR>\n" );
document.write( "For instance, intersect the line y = 40 and x+y = 60 to determine the location of point A(20,40)\r<BR>\n" );
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document.write( "After getting those corner points, we then will plug each into the profit function to see which yields the largest value of P.\r<BR>\n" );
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document.write( "If we tried the x and y coordinates of point A, then,<BR>\n" );
document.write( "P = 300x+150y<BR>\n" );
document.write( "P = 300*20+150*40<BR>\n" );
document.write( "P = 12000<BR>\n" );
document.write( "Repeat for points B through D\r<BR>\n" );
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document.write( "These are the results you should get<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Point</TD><TD >Coordinates</TD><TD >Profit (Php)</TD></tr><tr><TD >A</TD><TD >(20,40)</TD><TD >12000</TD></tr><tr><TD >B</TD><TD >(40,40)</TD><TD >18000</TD></tr><tr><TD >C</TD><TD >(50,30)</TD><TD >19500</TD></tr><tr><TD >D</TD><TD >(50,10)</TD><TD >16500</TD></tr>\n" );
document.write( "</TABLE>Point C is the winner in terms of max profit of Php19500<BR>\n" );
document.write( "The min profit happens at point A.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "How many beds on both types must be produced to maximize the profit? <font color=red>50 of type A and 30 of type B</font>\r<BR>\n" );
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document.write( "What is the minimum profit?  <font color=red>Php 12000</font>\r<BR>\n" );
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document.write( "Edit: just in case you made a typo and are asking for the maximum profit, then that max profit would be <font color=red>Php 19500</font><BR>\n" );
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