document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137721>Question 1137721</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A farmer can buy two types of plant​ food, mix A and mix B. Each cubic yard of mix A contains 52 pounds of phosphoric​ acid, 27 pounds of​ nitrogen, and 8 pounds of potash. Each cubic yard of mix B contains 13 pounds of phosphoric​ acid, 27 pounds of​ nitrogen, and 16 pounds of potash. The minimum monthly requirements are 520 pounds of phosphoric​ acid, 810 pounds of​ nitrogen, and 320 pounds of potash. Find the set of feasible solutions graphically for the amounts of mix A and mix B that can be used. If x is the number of cubic yards of mix A used and y is the number of cubic yards of mix B​ used, write a system of linear inequalities that indicates appropriate restraints on x and y. Write an inequality for the constraint on phosphoric acid </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755602 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=755602\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = number of cubic yards of mix A.<BR>\n" );
document.write( "y = number of cubic yards of mix B.\r<BR>\n" );
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document.write( "set up a table as follows:\r<BR>\n" );
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document.write( "                                  mix A      mix B    requirement\r\n" );
document.write( "number of cubic yards               x          y         >= 0\r\n" );
document.write( "pounds of phosphoric acid           52         13        >= 520\r\n" );
document.write( "pounds of nitrogen                  27         27        >= 810\r\n" );
document.write( "pounds of potash                    8          16        >= 320\r\n" );
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document.write( "your constraint equations are:\r<BR>\n" );
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document.write( "52x + 13y >= 520<BR>\n" );
document.write( "27x + 27y >= 810<BR>\n" );
document.write( "8x + 16y >= 320<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "using the decmos.com calculator, you wold graph the opposite of these inequalities.\r<BR>\n" );
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document.write( "the area of the graph that is not shaded is your region of feasibility.\r<BR>\n" );
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document.write( "the corner points of the feasibility region contain the maximum / minimum values for the objective function, if you have one.\r<BR>\n" );
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document.write( "the objective function coulde be maximize profit, minimize cost, or any other such business objective.\r<BR>\n" );
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document.write( "this problem doesn't have one, so we won't evaluate the objective function at each of the corner points.\r<BR>\n" );
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document.write( "we'll just graph the region of feasibility and identify the corner points.\r<BR>\n" );
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document.write( "here's what the graph looks like.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/033001.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the region of feasibility appears to be open ended because there is no limit on the maximum amount of ingredients that can be used; there is only a limit on the minimum amount of ingredients that can be used.\r<BR>\n" );
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document.write( "the corner points of the feasible region are (0, 53.33...), (40,0).\r<BR>\n" );
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document.write( "the choices are therefore, either all of mix A or all of mix B.\r<BR>\n" );
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document.write( "there doesn't appear to be a choice of some of mix A and some of mix B, as is normally seen with other type problems.\r<BR>\n" );
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document.write( "if the objective function was to minimize cost, then you would have a cost per cubic yard of mix ?A and a cost per cubic yard of mix B.\r<BR>\n" );
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document.write( "you would then evaluate the objective function at each of the corner points and select the corner point that has the minimum cost.\r<BR>\n" );
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document.write( "for example, if the cost for a cubic yard of mix A was 100 dollars and the cost for a cubic yard of mix B was 50 dollars, you would see that (0,53.33) would cost 50 * 53.33 and (40,0) would cost 100 * 40 and you would pick mix A because the cost for mix A would 2667 and the cost for mix B would be 4000.\r<BR>\n" );
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document.write( "if the cost for each mix were 100, then you would pick mix B because mix A would cost 5333 and mix B would cost 4000.\r<BR>\n" );
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document.write( "in a minimize / maximize type problem, you identify the corner points and then evaluate the objective function at those corner points.\r<BR>\n" );
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document.write( "your objective in this problem is to write a system of linear inequalities that indicates appropriate restraints on x and y. Write an inequality for the constraint on phosphoric acid.\r<BR>\n" );
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document.write( "that was done above.\r<BR>\n" );
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document.write( "the system of linear inequalities is:\r<BR>\n" );
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document.write( "52x + 13y >= 520<BR>\n" );
document.write( "27x + 27y >= 810<BR>\n" );
document.write( "8x + 16y >= 320<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "appropriate restraints on x and y are that they both have to be greater than or equal to 0.\r<BR>\n" );
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document.write( "linear inequality for phosphorous is 52x + 13y >= 520\r<BR>\n" );
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document.write( "note that the method used with the desmos.com calculator is not available with a lot of other graphing software.\r<BR>\n" );
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document.write( "most other software doesn't allow you to graph inequalities.\r<BR>\n" );
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document.write( "most other software requires you to transform the equation into slope intercept form of y = mx + b.\r<BR>\n" );
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document.write( "using the desmos.com software makes the job of graphing inequalities easy and will definitely spoil you once you learn how to use it.\r<BR>\n" );
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document.write( "the technique of graphing the opposite of the inequalities is recommended because it's much easier to spot the feasibility region than graphing the inequalities as is.\r<BR>\n" );
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document.write( "with a lot of other software, you need to convert the inequalities to y = mx +  b form and graph the equality portion of the inequalities  and then manually shade the area of the graph that satisfies the inequalities.\r<BR>\n" );
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document.write( "there's a lot more manual work involved that is eliminated through the use of the desmos.com calculator.\r<BR>\n" );
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document.write( "if you cannot use the desmos.com calculator, or have to graph using some other software, then come back to me and i'll show you how to do it.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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