document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117091>Question 1117091</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is Linear Programming Pls help us! :( <BR>\n" );
document.write( "please use 2 variables only x and y thank you <BR>\n" );
document.write( "MILESTONE: Linear Program <BR>\n" );
document.write( "After making various presentations to potential investors, MC has finally hit the jackpot. An investor wants to invest 100,000,000 pesos in MC. <BR>\n" );
document.write( "After doing all your staff work in the previous weeks, the two of you have been tasked to find the most efficient way to invest the money. This involves making the highest profit possible from the money you have invested. You have decided to consult your friend Olivia, who is also a financial investor. <BR>\n" );
document.write( "Olivia discusses that you need to use a mathematical method called Linear Programming to solve this problem. She explains that Linear Programming is a method that has to be set up very carefully. It involves creating a function to optimize and modelling various constraints using linear equations. She has decided to present you the profitability percentages of MC, given that she has already done prior work with MC. <BR>\n" );
document.write( "The startup arm is projected to return at least 12%. To minimize risk, you must invest no more than 30,000,000 pesos. The financial stocks return 3%, while the retail arm returns 5%. <BR>\n" );
document.write( "For tax reasons, you must invest at least 3 times more in the financial stocks than the retail arm. <BR>\n" );
document.write( "Your task is to set up the linear program. You have to define the following:<BR>\n" );
document.write( "1.	The variables that will be used<BR>\n" );
document.write( "2.	The optimization function that will be used<BR>\n" );
document.write( "a.	What kind of optimization will be done: minimization or maximization?<BR>\n" );
document.write( "3.	The constraints that the linear program will be subjected to </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732089 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=732089\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I already answered this question recently (hopefully from a student unknown to you) as question number 1116231.<BR>\n" );
document.write( "A copy of my previous answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%22%28edited%29%22 ALIGN=MIDDLE  ALT=\"red%22%28edited%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is attached below.<BR>\n" );
document.write( "NOTES: Sometimes you can locate an answer to the same question using a phrase from the question in an internet search.<BR>\n" );
document.write( "If you do not like my answer to a question and are going to re-post it for a second opinion, just let me know through a thank you note, and I will leave the question for someone else to answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE PROBLEM:<BR>\n" );
document.write( "If I understand the story, there are 3 different investments being considered:<BR>\n" );
document.write( "A) a financial stocks investment that returns 3%,<BR>\n" );
document.write( "B) a slightly more risky retail company investment that returns 5%, and <BR>\n" );
document.write( "C) a new, startup company that is expected to return 12%, but is the riskiest choice.<BR>\n" );
document.write( "Because of the risk, you want to invest no more than 30,000,000, pesos in option C),<BR>\n" );
document.write( "and to balance risk between options A) and B),you want to invest at least 3 more times in A) than in B).<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE EXPECTED ANSWER:<BR>\n" );
document.write( "1. The variables that will be used<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= amount invested in B), the retail company, in pesos.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= amount invested in A), the financial stocks, in pesos.<BR>\n" );
document.write( "2. The optimization function that will be used<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%2Cy%29 ALIGN=MIDDLE  ALT=\"P%28x%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the profit returned by the mix of investments.<BR>\n" );
document.write( "To figure out what the function \"formula\" is,<BR>\n" );
document.write( "you multiply each amount invested times the percentage return expected (as a decimal),<BR>\n" );
document.write( "and add the results.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%225%25%22=5%2F100=0.05 ALIGN=MIDDLE  ALT=\"%225%25%22=5%2F100=0.05\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.05x ALIGN=MIDDLE  ALT=\"0.05x\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%223%25%22=3%2F100=0.03 ALIGN=MIDDLE  ALT=\"%223%25%22=3%2F100=0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.03y ALIGN=MIDDLE  ALT=\"0.03y\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos.<BR>\n" );
document.write( "The amount to be invested on C), the \"startup arm\" is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22100%2C000%2C000%22-x-y ALIGN=MIDDLE  ALT=\"%22100%2C000%2C000%22-x-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2212%25%22=12%2F100=0.12 ALIGN=MIDDLE  ALT=\"%2212%25%22=12%2F100=0.12\"   DISABLED_style= \"max-width:90%; height:auto;\" > profit expected from that investment is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.12%28%22100%2C000%2C000%22-x-y%29=%2212%2C000%2C000%22-0.12x-0.12y ALIGN=MIDDLE  ALT=\"0.12%28%22100%2C000%2C000%22-x-y%29=%2212%2C000%2C000%22-0.12x-0.12y\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%2Cy%29=0.05x%2B0.03y%2B%2212%2C000%2C000%22-0.12x-0.12y ALIGN=MIDDLE  ALT=\"P%28x%2Cy%29=0.05x%2B0.03y%2B%2212%2C000%2C000%22-0.12x-0.12y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%28x%2Cy%29=%2212%2C000%2C000%22-0.07x-0.09y%29 ALIGN=MIDDLE  ALT=\"highlight%28P%28x%2Cy%29=%2212%2C000%2C000%22-0.07x-0.09y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2a. What kind of optimization will be done: minimization or maximization?<BR>\n" );
document.write( "Do we want to maximize profit or to keep it to a minimum?<BR>\n" );
document.write( "We want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28maximization%29 ALIGN=MIDDLE  ALT=\"highlight%28maximization%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , of course.<BR>\n" );
document.write( "3. The constraints that the linear program will be subjected to<BR>\n" );
document.write( "We do not invest negative amounts, and that gives us the constraints<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%3E=0%29 ALIGN=MIDDLE  ALT=\"highlight%28x%3E=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y%3E=0%29 ALIGN=MIDDLE  ALT=\"highlight%28y%3E=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22100%2C000%2C000%22-x-y%3E=0 ALIGN=MIDDLE  ALT=\"%22100%2C000%2C000%22-x-y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28red%28x%2By%3C=%22100%2C000%2C000%22%29%29 ALIGN=MIDDLE  ALT=\"highlight%28red%28x%2By%3C=%22100%2C000%2C000%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We are told not to invest more than 30,000,000 in risky option C, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22100%2C000%2C000%22-x-y%3C=%2230%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"%22100%2C000%2C000%22-x-y%3C=%2230%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28green%28x%2By%3E=%2270%2C000%2C000%22%29%29 ALIGN=MIDDLE  ALT=\"highlight%28green%28x%2By%3E=%2270%2C000%2C000%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is another \"constraint\".<BR>\n" );
document.write( "The advice to invest \"at least 3 times more in the financial stocks than the retail arm\" translates into the constraint<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28blue%28y%3E=3x%29%29 ALIGN=MIDDLE  ALT=\"highlight%28blue%28y%3E=3x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "4. Although you did not ask, you would graph the \"feasible region\" determined by your constraints as the quadrilateral ABCD below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-20000000%2C120000000%2C-20000000%2C120000000%2C%0D%0Aline%28-30000000%2C0%2C120000000%2C0%29%2Carrow%28100000000%2C0%2C120000000%2C0%29%2C%0D%0Aline%280%2C-30000000%2C0%2C120000000%29%2Carrow%280%2C100000000%2C0%2C120000000%29%2C%0D%0Aline%2825000000%2C-2000000%2C25000000%2C2000000%29%2C%0D%0Aline%2850000000%2C-2000000%2C50000000%2C2000000%29%2C%0D%0Aline%2875000000%2C-2000000%2C75000000%2C2000000%29%2C%0D%0Aline%28100000000%2C-2000000%2C100000000%2C2000000%29%2C%0D%0Aline%28-2000000%2C25000000%2C2000000%2C25000000%29%2C%0D%0Aline%28-2000000%2C50000000%2C2000000%2C50000000%29%2C%0D%0Aline%28-2000000%2C75000000%2C2000000%2C75000000%29%2C%0D%0Aline%28-2000000%2C100000000%2C2000000%2C100000000%29%2C%0D%0Alocate%2885000000%2C-2500000%2C%22100%2C000%2C000%22%29%2C%0D%0Alocate%28-20000000%2C105000000%2C%22100%2C000%2C000%22%29%2C%0D%0Ared%28line%28100000000%2C0%2C0%2C100000000%29%29%2C%0D%0Agreen%28line%2870000000%2C0%2C0%2C70000000%29%29%2C%0D%0Ablue%28line%280%2C0%2C50000000%2C150000000%29%29%2C%0D%0Alocate%28-4000000%2C72000000%2CD%29%2Clocate%28-4000000%2C100000000%2CC%29%2C%0D%0Alocate%2817000000%2C50000000%2CA%29%2Clocate%2827000000%2C78000000%2CB%29%2C%0D%0Alocate%28118000000%2C0%2Cx%29%2Clocate%282000000%2C120000000%2Cy%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The blue line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y=3x%29 ALIGN=MIDDLE  ALT=\"blue%28y=3x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . The points on that line and those above the line satisfy constraint <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28y%3E=3x%29 ALIGN=MIDDLE  ALT=\"blue%28y%3E=3x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The red line, with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%2By=%22100%2C000%2C000%22%29 ALIGN=MIDDLE  ALT=\"red%28x%2By=%22100%2C000%2C000%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28added%29 ALIGN=MIDDLE  ALT=\"red%28added%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the points below it satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%2By%3C=%22100%2C000%2C000%22%29 ALIGN=MIDDLE  ALT=\"red%28x%2By%3C=%22100%2C000%2C000%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The green line, with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28x%2By=%2270%2C000%2C000%22%29 ALIGN=MIDDLE  ALT=\"green%28x%2By=%2270%2C000%2C000%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28added%29 ALIGN=MIDDLE  ALT=\"red%28added%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and the points above it satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28x%2By%3E=%2270%2C000%2C000%22%29 ALIGN=MIDDLE  ALT=\"green%28x%2By%3E=%2270%2C000%2C000%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The points in the first quadrant, including the origin and the positive x- and y-axes satisfy the constraints <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The points inside and on the border of quadrilateral ABCD satisfy all constraints and are the feasible region.<BR>\n" );
document.write( "5. Although you did not ask, each profit value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%2Cy%29=K ALIGN=MIDDLE  ALT=\"P%28x%2Cy%29=K\"   DISABLED_style= \"max-width:90%; height:auto;\" > graphs as a staring line.<BR>\n" );
document.write( "For example,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%225%2C700%2C000%22=%2212%2C000%2C000%22-0.07x-0.09y ALIGN=MIDDLE  ALT=\"%225%2C700%2C000%22=%2212%2C000%2C000%22-0.07x-0.09y\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.07x%2B0.09y=%226%2C300%2C000%22 ALIGN=MIDDLE  ALT=\"0.07x%2B0.09y=%226%2C300%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%228%2C850%2C000%22=%2212%2C000%2C000%22-0.07x-0.09y ALIGN=MIDDLE  ALT=\"%228%2C850%2C000%22=%2212%2C000%2C000%22-0.07x-0.09y\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.07x%2B0.09y=%223%2C150%2C000%22 ALIGN=MIDDLE  ALT=\"0.07x%2B0.09y=%223%2C150%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "graph as the black slanted lines shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-20000000%2C120000000%2C-20000000%2C120000000%2C%0D%0Aline%28-30000000%2C0%2C120000000%2C0%29%2Carrow%28100000000%2C0%2C120000000%2C0%29%2C%0D%0Aline%280%2C-30000000%2C0%2C120000000%29%2Carrow%280%2C100000000%2C0%2C120000000%29%2C%0D%0Aline%2825000000%2C-2000000%2C25000000%2C2000000%29%2C%0D%0Aline%2850000000%2C-2000000%2C50000000%2C2000000%29%2C%0D%0Aline%2875000000%2C-2000000%2C75000000%2C2000000%29%2C%0D%0Aline%28100000000%2C-2000000%2C100000000%2C2000000%29%2C%0D%0Aline%28-2000000%2C25000000%2C2000000%2C25000000%29%2C%0D%0Aline%28-2000000%2C50000000%2C2000000%2C50000000%29%2C%0D%0Aline%28-2000000%2C75000000%2C2000000%2C75000000%29%2C%0D%0Aline%28-2000000%2C100000000%2C2000000%2C100000000%29%2C%0D%0Alocate%2885000000%2C-2500000%2C%22100%2C000%2C000%22%29%2C%0D%0Alocate%28-20000000%2C105000000%2C%22100%2C000%2C000%22%29%2C%0D%0Ared%28line%28100000000%2C0%2C0%2C100000000%29%29%2C%0D%0Agreen%28line%2870000000%2C0%2C0%2C70000000%29%29%2C%0D%0Ablue%28line%280%2C0%2C50000000%2C150000000%29%29%2C%0D%0Alocate%28-4000000%2C72000000%2CD%29%2Clocate%28-4000000%2C100000000%2CC%29%2C%0D%0Alocate%2817000000%2C50000000%2CA%29%2Clocate%2827000000%2C78000000%2CB%29%2C%0D%0Alocate%28118000000%2C0%2Cx%29%2Clocate%282000000%2C120000000%2Cy%29%2C%0D%0Aline%280%2C70000000%2C90000000%2C0%29%2Cline%280%2C35000000%2C45000000%2C0%29%2C%0D%0Alocate%2838000000%2C12000000%2C0.07x%2B0.09y=%223%2C150%2C000%22%29%2C%0D%0Alocate%2835000000%2C50000000%2C0.07x%2B0.09y=%226%2C300%2C000%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "All such lines have the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7%2F9 ALIGN=MIDDLE  ALT=\"-7%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > slope, so they are parallel,<BR>\n" );
document.write( "with a profit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%226%2C300%2C000%22 ALIGN=MIDDLE  ALT=\"%226%2C300%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos represented by the upper line shown, and a profit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%228%2C850%2C000%22 ALIGN=MIDDLE  ALT=\"%228%2C850%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos represented by the lower line.<BR>\n" );
document.write( "As the profit increases the line slides down, parallel to itself,<BR>\n" );
document.write( "but the lower line is outside the feasible region,<BR>\n" );
document.write( "failing some of the constraints.<BR>\n" );
document.write( "It is obvious that in this case, the maximum profit will happen at point A,<BR>\n" );
document.write( "as a fifth grader could tell you without much math.<BR>\n" );
document.write( "With this type of problem, the lines for constant optimum value slide until they get to a corner or an edge of the feasible region.<BR>\n" );
document.write( "That is why your teacher will tell you to calculate the value of the optimization function at all corners of the feasible region to find the optimum value of the function and the conditions (x and y) that produce that optimum value.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28%22EDIT%22%29 ALIGN=MIDDLE  ALT=\"red%28%22EDIT%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (added calculations):<BR>\n" );
document.write( "For point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the coordinates are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=%2225%2C000%2C000%22%2Cy=%2275%2C000%2C000%22%29 ALIGN=MIDDLE  ALT=\"system%28x=%2225%2C000%2C000%22%2Cy=%2275%2C000%2C000%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the solution to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28y=3x%29%2Cred%28x%2By=%22100%2C000%2C000%22%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28y=3x%29%2Cred%28x%2By=%22100%2C000%2C000%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That means investing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2225%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"%2225%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos on the retail company,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2275%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"%2275%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos on financial stocks, and nothing on the startup.<BR>\n" );
document.write( "and the value of the optimization function is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28%2225%2C000%2C000%22%2C%2225%2C000%2C000%22%29=%2212%2C000%2C000%22-0.07%2A%2225%2C000%2C000%22-0.09%2A%2225%2C000%2C000%22=%223%2C500%2C000%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "meaning a profit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%223%2C500%2C000%22 ALIGN=MIDDLE  ALT=\"%223%2C500%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos.<BR>\n" );
document.write( "For point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the coordinates are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=%2217%2C500%2C000%22%2Cy=%2252%2C500%2C000%22%29 ALIGN=MIDDLE  ALT=\"system%28x=%2217%2C500%2C000%22%2Cy=%2252%2C500%2C000%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the solution to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28blue%28y=3x%29%2Cgreen%28x%2By=%2270%2C000%2C000%22%29%29 ALIGN=MIDDLE  ALT=\"system%28blue%28y=3x%29%2Cgreen%28x%2By=%2270%2C000%2C000%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That means investing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2217%2C500%2C000%22 ALIGN=MIDDLE  ALT=\"%2217%2C500%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos on the retail company,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2252%2C500%2C000%22 ALIGN=MIDDLE  ALT=\"%2252%2C500%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos on financial stocks, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2230%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"%2230%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >  on the startup.<BR>\n" );
document.write( "and the value of the optimization function is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28%2217%2C500%2C000%22%2C%2252%2C500%2C000%22%29=%2212%2C000%2C000%22-0.07%2A%2217%2C500%2C000%22-0.09%2A%2252%2C500%2C000%22=%226%2C050%2C000%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you choose point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%22100%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"y=%22100%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you are investing all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22100%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"%22100%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > on financial stocks,<BR>\n" );
document.write( "for a profit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%2C%22100%2C000%2C000%22%29=%2212%2C000%2C000%22-0.07%2A0-0.09%2A%22100%2C000%2C000%22=%223%2C000%2C000%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > pesos.<BR>\n" );
document.write( "If you choose point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2270%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"y=%2270%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you are investing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2270%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"%2270%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos on financial stocks,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2270%2C000%2C000%22 ALIGN=MIDDLE  ALT=\"%2270%2C000%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pesos on the startup,<BR>\n" );
document.write( "for a profit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%2C%22100%2C000%2C000%22%29=%2212%2C000%2C000%22-0.07%2A0-0.09%2A%2270%2C000%2C000%22=%225%2C700%2C000%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > pesos.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE FIFTH-GRADER SAYS:<BR>\n" );
document.write( "Obviously, you want to invest as much as possible where you get the most profit,<BR>\n" );
document.write( "so you should be aggressive and invest 30,000,000, pesos in option C).<BR>\n" );
document.write( "For the remaining 100,000,000-30,000,000=70,000,000 pesos,<BR>\n" );
document.write( "you want to invest as much as possible in B), <BR>\n" );
document.write( "but since they advise to invest  at least 3 more times in A) than in B),<BR>\n" );
document.write( "you split that 70,000,000 pesos into four equal parts,<BR>\n" );
document.write( "invest 1/4 (17,500,500 pesos) in B),<BR>\n" );
document.write( "and the other 3/4 (52,500,000 pesos in A).<BR>\n" );
document.write( "You expect to earn<BR>\n" );
document.write( "12% of 30,000,000 pesos (3,600,000 pesos) from C),<BR>\n" );
document.write( "5% of 17,500,000 pesos (875,000 pesos from B), and<BR>\n" );
document.write( "3% of 52,500,000 pesos (1,575,000 pesos from A),<BR>\n" );
document.write( "for a total return of 6,050,000 pesos<BR>\n" );
document.write( "(6.05% of your 100,000,000 investment),<BR>\n" );
document.write( "which is the maximum profit predicted.<BR>\n" );
document.write( "End of story.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE MORE IMPRESSIVE EXPECTED GROWN-UP RESPONSE:<BR>\n" );
document.write( "You produce a slide presentation with lots of color, music, and inspirational pictures.<BR>\n" );
document.write( "You show tables of data with marketing studies' results backing up the projected 12% return for the new startup company.<BR>\n" );
document.write( "You present expert opinions backing up the investment constraints advice.<BR>\n" );
document.write( "You throw in jargon financial terms,<BR>\n" );
document.write( "mention linear programming as an important mathematical tool,<BR>\n" );
document.write( "and present the answers to the problem questions,<BR>\n" );
document.write( "with all possible mathematical terms and graphs.<BR>\n" );
document.write( "Your mesmerized audience congratulates you on your analysis on how to maximize profits.<BR>\n" );
document.write( "You get a promotion,<BR>\n" );
document.write( "and the fifth grader maximizes the number of chocolate chip cookies that can be swiped from the board room table. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );