document.write( "Question 1185661: A factory manufactures chairs and tables, each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 39 hours; the second at most 42 hours; and the third at most 23 hours. A chair requires 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; a table needs 2 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is $20 per unit for a chair and $30 for a table, how many units of each should be manufactured to maximize profit? \n" ); document.write( "
Algebra.Com's Answer #816523 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
let x = the number of chairs and y = the number of tables.
\n" ); document.write( "the chair requires 1 hours of cutting, 2 hours of assembly, and 1 hour of finishing.
\n" ); document.write( "the table needs 2 hours of cutting, 1 hour of assembly, and 1 hour of finishing.
\n" ); document.write( "the maximum hours of cutting = 39 hours.
\n" ); document.write( "the maximum hours of assembly = 42 hours.
\n" ); document.write( "the maximum hours of finishing = 23 hours.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "make a table as shown below:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "                             chair              table\r\n" );
document.write( "number of                      x                  y\r\n" );
document.write( "cutting hours                  1                  2           <= 39 hours\r\n" );
document.write( "assembly hours                 2                  1           <= 42 hours\r\n" );
document.write( "finishing hours                1                  1           <= 23 hours\r\n" );
document.write( "\r\n" );
document.write( "profit                         20                 30          maximize\r\n" );
document.write( "\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "your objective function is:
\n" ); document.write( "profit = 20x + 30y.
\n" ); document.write( "this is what you want to maximize.\r
\n" ); document.write( "\n" ); document.write( "your constraint functions are:
\n" ); document.write( "x + 2y <= 39
\n" ); document.write( "2x + y <= 42
\n" ); document.write( "x + y <= 23\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "using the desmos.com calculator, you would:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "graph the opposite of the inequalities.
\n" ); document.write( "the maximum profit will be at the corners of the feasible region, which is the unshaded area of the graph.
\n" ); document.write( "evaluate the objective function at each ot these corner points.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the maximum profit is at the coordinate point of (7,16).
\n" ); document.write( "profit = 20*7 + 30*16 = 620
\n" ); document.write( "cutting hours = 1*7 + 2*16 = 39 <= 39
\n" ); document.write( "assembly hours = 2*7 + 1*16 = 30 <= 42
\n" ); document.write( "finishing hours = 1*7 + 1*16 = 23 >= 23\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "your solution is that he should make 7 chairs and 16 tables to maximize his profit and still be within the limits of the constraints.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the graphical solution looks like this:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );