Algebra.Com's Answer #700838 by ikleyn(52832)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "find the maximum value of p=5x+3y under the following constraints:2x+y=<20, 2x+3y=<24, x=>0 and y=>o
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "I will solve it by the linear programming method.
\n" );
document.write( "I will not explain here how the method works in general case - find other sourses for it.\r
\n" );
document.write( "\n" );
document.write( "I will only explain how it works for the given concrete problem.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "1. Make a plot of the feasible area.\r\n" );
document.write( " It is restricted by the straight lines \r\n" );
document.write( " \r\n" );
document.write( " 2x + y = 20, or y = 20 - 2x;\r\n" );
document.write( " 2x + 3y = 24, or y =  ;\r\n" );
document.write( " x = 0 (y-axis) and\r\n" );
document.write( " y = 0 (x-axis).\r\n" );
document.write( "\r\n" );
document.write( " See the plot below:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " Feasibility area. Straight lines 2x + y = 20 (red); 2x + 3y = 24 (green)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " Feasibility area is this quadrilateral in the first quadrant restricted by the straight lines and axes.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2. The critical points are the vertices of this quadrilateral.\r\n" );
document.write( "\r\n" );
document.write( " One vertex is x-intercept of the red line. It is the point (10,0).\r\n" );
document.write( "\r\n" );
document.write( " Second vertex is y-intercept of the green line. It is the point (0,8).\r\n" );
document.write( "\r\n" );
document.write( " Third vertex is the intersection point of the red and the green line.\r\n" );
document.write( " To find the coordinates of this point, you need to solve the system of two linear equations.\r\n" );
document.write( " I did it for you and found the coordinates (x,y) = (9,2).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3. The last step you need to do is to calculate the values of your \"profit\" function P(x,y) = 5x + 3y at vertices. You have\r\n" );
document.write( "\r\n" );
document.write( " First vertex: P(10,0) = 5*10 + 3*0 = 50.\r\n" );
document.write( "\r\n" );
document.write( " Second vertex: P(0,8) = 5*0 + 3*8 = 24.\r\n" );
document.write( "\r\n" );
document.write( " Third vertex: P(9,2) = 5*9 + 3*2 = 45 + 6 = 51.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "4. Your \"profit\" function is maximal at the third vertex.\r\n" );
document.write( " \r\n" );
document.write( " Then the linear programming method states that it is the solution of your minimax problem.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " The function p(x,y) = 5x + 3y has the maximum value over the feasibility area at the point (x,y) = (9,2), \r\n" );
document.write( " and this maximum value is equal to 51.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "The solution is completed. The problem is solved.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "See also the lesson\r
\n" );
document.write( "\n" );
document.write( " - Solving minimax problems by the Linear Programming method \r
\n" );
document.write( "\n" );
document.write( "in this site.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |