document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151594>Question 1151594</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You make decorative stones for landscaping. A ton of coarse stones requires 2 hours of crushing, 5 hours of sifting, and 8 hours of drying. A ton of fine stones requires 6 hours of crushing, 3 hours of sifting, and 2 hours of drying. The coarse stones sell for $400 per ton. The fine stones sell for $500 per ton. In a work week your plant is capable of 36 hours of crushing, 30 hours of sifting, and 40 hours of drying.<BR>\n" );
document.write( "Use corner solution, to Determine:<BR>\n" );
document.write( "a. How much of each kind of stones you should make to maximize your revenue.<BR>\n" );
document.write( "b. How much revenue you'll make at the maximum.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773396 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=773396\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = number of tons of coarse stones.<BR>\n" );
document.write( "y = number of tons of fine stones.<BR>\n" );
document.write( "make a table as shown below:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "description              coarse stones        fine stones\r\n" );
document.write( "number of tons                 x                   y\r\n" );
document.write( "crushing                       2                   6          <= 36\r\n" );
document.write( "sifting                        5                   3          <= 30\r\n" );
document.write( "drying                         8                   2          <= 40\r\n" );
document.write( "revenue                        400                 500        maximize\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "your objective function is:<BR>\n" );
document.write( "400x + 500y<BR>\n" );
document.write( "this is what your want to maximize.<BR>\n" );
document.write( "your constraint functions are:<BR>\n" );
document.write( "2x + 6y <= 36<BR>\n" );
document.write( "5x + 3y <= 30<BR>\n" );
document.write( "8x + 2y <= 40<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0<BR>\n" );
document.write( "using the decmos.com calculator, you would graph the opposite of the constraints<BR>\n" );
document.write( "for example:<BR>\n" );
document.write( "2x + 6y <= 36 is graphed as 2x + 6y >= 36<BR>\n" );
document.write( "the area of the graph that is NOT shaded is the region of feasibility.<BR>\n" );
document.write( "the maximum revenue will be at the corner points of this region.\r<BR>\n" );
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document.write( "the equations used for graphing are shown below\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/012501.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "the graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/012502.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "the objective function is evaluated at each corner point.<BR>\n" );
document.write( "the corner point with the maximum revenue is the solution.\r<BR>\n" );
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document.write( "(0,6) becomes 400 * 0 + 500 * 6 = 3000<BR>\n" );
document.write( "(3,5) becomes 400 * 3 + 500 * 5 = 3700<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "the maximum revenue is found at (3,5).\r<BR>\n" );
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document.write( "all the constraints have to be satisfied.<BR>\n" );
document.write( "this is true for (3,5) so that solution appears to be good.\r<BR>\n" );
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document.write( "some adjustment might need to be made at the point with fractions in it if only integer tons can be sold.<BR>\n" );
document.write( "that would be (4.286, 2.857).<BR>\n" );
document.write( "the options would be:<BR>\n" );
document.write( "(4,2)<BR>\n" );
document.write( "4,3)<BR>\n" );
document.write( "(5,2)<BR>\n" );
document.write( "(5,3)<BR>\n" );
document.write( "the only one of those points where all the constraints were met was (4,2) and that point did not produce maximum revenue.<BR>\n" );
document.write( "this analysis might be moot since (4.286,2.857) did not provide maximum reve nue, but it's still a consideration that might need to be explored.\r<BR>\n" );
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document.write( "based on the graphical analysis using desmos.com calculator, the maximum revenue is when 3 tons of coarse stones and 5 tons of fine stones are sold.\r<BR>\n" );
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document.write( "there is a simplex method tool that can also be used.<BR>\n" );
document.write( "graphical solutionis only good for 2 variables.<BR>\n" );
document.write( "simplex method tool can be used for more than 2 variables.<BR>\n" );
document.write( "this tool can be found at:<BR>\n" );
document.write( "https://www.zweigmedia.com/RealWorld/simplex.html<BR>\n" );
document.write( "i used this tool to see if it came up with the same solution as the graphical solution.<BR>\n" );
document.write( "it did.<BR>\n" );
document.write( "using this tool, you provide the constraints as is and you do not provide the opposite of the constraint inequalities.<BR>\n" );
document.write( "the results are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/012503.jpg\" alt=\"$$$\" >\r<BR>\n" );
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