document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177067>Question 1177067</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pension fund manager decides to invest a total of at most ​$40 million in U.S. Treasury bonds paying 3​% annual interest and in mutual funds paying 5% <BR>\n" );
document.write( "annual interest. He plans to invest at least ​$5 million in bonds and at least ​$20 million in mutual funds. Bonds have an initial fee of​ $100 per million​ dollars, while the fee for mutual funds is​ $200 per million. The fund manager is allowed to spend no more than ​$7000 on fees. How much should be invested in each to maximize annual​ interest? What is the maximum annual interest? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #805876 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=805876\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = the amount invested in bonds.<BR>\n" );
document.write( "let y the amount invested in mutual funds.\r<BR>\n" );
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document.write( "everything will be converted to millions for ease of presentation.\r<BR>\n" );
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document.write( "your constraint functions are:\r<BR>\n" );
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document.write( "x + y <= 40<BR>\n" );
document.write( "x >= 5 <BR>\n" );
document.write( "y >= 20 <BR>\n" );
document.write( ".0001 * x + .0002 * y <= .007<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "your objective function is\r<BR>\n" );
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document.write( ".03 * x + .05 * y\r<BR>\n" );
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document.write( "this is what you want to maximize.\r<BR>\n" );
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document.write( "using the desmos.com calculator, you would graph the opposite of these constraint functions.\r<BR>\n" );
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document.write( "the coordinate points on the area of the graph that is not shaded, and the coordinate points on the lines themselves, if the inequalities are expressed as <= or >=, will be in the feasible region.\r<BR>\n" );
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document.write( "if the inequalities are expressed as < or >, then the coordinate points on the lines themselves are not included in the feasible region.\r<BR>\n" );
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document.write( "the graph looks like this.\r<BR>\n" );
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document.write( "you would evaluate the objective function at each of the corner points of the feasible region on the graph.\r<BR>\n" );
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document.write( "the coordinate points that contain the maximum value for the objective function is your maximum solution.\r<BR>\n" );
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document.write( "i created a chart that showed the following:\r<BR>\n" );
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document.write( "a               b           c           d\r\n" );
document.write( "(5,32.5)        1.775       37.5        .007\r\n" );
document.write( "(10,30)         1.8         40          .007\r\n" );
document.write( "(5,20)          1.15        25          .0045\r\n" );
document.write( "(20,20)         1.6         40          .006\r\n" );
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document.write( "column a is the coordinate points in (x,y) format.<BR>\n" );
document.write( "x represents the amount invested in bonds.<BR>\n" );
document.write( "y represents the amount invested in mutual funds.\r<BR>\n" );
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document.write( "column b = .03 * x + .05 * y<BR>\n" );
document.write( "this is the profit.\r<BR>\n" );
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document.write( "column c = x + y<BR>\n" );
document.write( "this is the total amount invested.\r<BR>\n" );
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document.write( "column d = .0001 * x + .0002 * y<BR>\n" );
document.write( "this is the total amount paid out in fees.\r<BR>\n" );
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document.write( "from the table, it's clear that all of the constraints have been met for all of the options at the corner point of the feasible region.\r<BR>\n" );
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document.write( "the coordinate points with the maxim profit are at (x,y) = (10,30).<BR>\n" );
document.write( "this means 10 million invested in bonds and 30 million invested in mutual funds.<BR>\n" );
document.write( "you can see that the profit is 1.8 million with 40 million invested and 7000 paid out in fees (.007 million is equal to 7000).\r<BR>\n" );
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document.write( "the maximum interest is 1.8 million with 10 million invested in bonds and 30 million invested in mutual funds.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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