document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191606>Question 1191606</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer produces a business calculator and a graphing calculator. each calculator is assembled in two sets of operations, where each operation is in production 8 hours during each day.  the average time required for a business calculatorin the first operation is 3 minutes, and 6 minutes is required in the second operation. the graphing calculator averages 6 minutes in the first operation and 4 minutes in the second operation. all calculators can be sold the profit for business calculator is 8$ and the profit for a graphing calculatoris $10.  how many of each type of calculator should be made each day in order to maximize profit.\r<BR>\n" );
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document.write( "I know that one equation will include X= the business calculator and y= the graphing calculator.  <BR>\n" );
document.write( "x+y<=8 hours?<BR>\n" );
document.write( "9x+10y<=? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823397 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823397\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i think you need to convert hours to minutes, since the operations are in minutes.\r<BR>\n" );
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document.write( "alternatively, you can convert minutes to hours, but it seems to make more sense to convert hours to minutes.<BR>\n" );
document.write( "8 hours * 60 minutes per hour = 480 minutes.\r<BR>\n" );
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document.write( "let x = the number of business calculators.<BR>\n" );
document.write( "let y = the number of graphing calculators.\r<BR>\n" );
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document.write( "since the profit per business calculator is 8 dollars and the profit per graphing calculator is 10, then your objective function is profit = 8x + 10y.\r<BR>\n" );
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document.write( "this is the equation that will be used to evaluate the corner points of the feasible region.\r<BR>\n" );
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document.write( "you have two operations required to assemble a calculator.\r<BR>\n" );
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document.write( "for the business calculator, the first operation requires 3 minutes and the second operation requires 6 minutes.\r<BR>\n" );
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document.write( "for the graphing calculator, the first operation requires 6 minutes and the second operation requires 4 minutes.\r<BR>\n" );
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document.write( "each of these operations can be used for up to 8 hours each day.<BR>\n" );
document.write( "translated to minutes, each of those operations can be used for up to 480 minutes each day.\r<BR>\n" );
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document.write( "your constraint inequalities are:\r<BR>\n" );
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document.write( "3x + 6y <= 480<BR>\n" );
document.write( "6x + 4y <= 480\r<BR>\n" );
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document.write( "the first inequality says that 3 minutes times the number of business calculators and 6 minutes times the number of graphing calculators must total to less than or equal to 480 minutes each day for the first operation.\r<BR>\n" );
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document.write( "the second inequality says that 6 minutes times the  number of business calculators and 4 minutes times the number of graphing calculators must total to less than or equal to 480 minutes each day for the second operation.\r<BR>\n" );
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document.write( "two additional constraints are that the number of business calculators and the number of graphing calculator can't be negative.\r<BR>\n" );
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document.write( "those inequalities are:<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "using the desmos.com calculator, you would graph the opposite of these inequalities.<BR>\n" );
document.write( "the area on the graph that is not shaded will be your region of feasibility.<BR>\n" );
document.write( "the lines of the equations themselves will also be in the region of feasibility if the inequalities are <= or >=, rather than < or >.\r<BR>\n" );
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document.write( "to summarize.\r<BR>\n" );
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document.write( "your objective function is:<BR>\n" );
document.write( "8x + 106 = profit<BR>\n" );
document.write( "your constraint inequalities are:<BR>\n" );
document.write( "3x + 6y <= 480<BR>\n" );
document.write( "6x + 4y <= 480<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "you are graphing the constraints and you are evaluating each corner point with the objective function.\r<BR>\n" );
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document.write( "the graph is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2022/030301.jpg\" >\r<BR>\n" );
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document.write( "the corner points on the graph are (0,80), (40,60), (80,0).<BR>\n" );
document.write( "these are coordinate points in (x,y) format.<BR>\n" );
document.write( "x is the number of business calculators.<BR>\n" );
document.write( "y is the number of graphing calculators.\r<BR>\n" );
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document.write( "you would evaluate the objective function of profit = 8x + 10y at each of the corner points of the feasible region.\r<BR>\n" );
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document.write( "the corner point with the maximum profit will be at (40,60).\r<BR>\n" );
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document.write( "all the constraints need to be also satisfied, not just some.<BR>\n" );
document.write( "3x + 6y <= 480 becomes 120 + 360 <= 480 which becomes 480 <= 480 which is true.<BR>\n" );
document.write( "6x + 4y  480 becomes 240 + 240 <= 480 which becomes 480 <= 480 which is also true.<BR>\n" );
document.write( "both x and y are greater than or equal to 0.\r<BR>\n" );
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document.write( "since all the constraints are satisfied at the maximum profit point, then the maximum profit will be when 40 business calculator and 60 graphing calculators are assembled and sold.\r<BR>\n" );
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document.write( "i'll be available to answer any questions or concerns you have about this analysis.<BR>\n" );
document.write( "theo\r<BR>\n" );
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document.write( " </SPAN>\n" );
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