document.write( "Question 1123606: The manufacturer of dolls want to maximize their revenue for the sale of the dolls. Each girl doll sold yields $9 in revenue and each boy doll yields $9.50. The dolls are made of plastic and nylon. To make a girl doll requires 12 oz of plastic and 5 oz of nylon. A boy doll requires 14oz of plastic but no nylon. The manufacturer has no more than 100,000 oz of plastic and 30,000o of nylon available each week. Kids will buy at least twice as many girl dolls as boys. How many of each should they manufacture each week to maximize their revenue? \n" ); document.write( "

Algebra.Com's Answer #739969 by ikleyn(52798) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Unfortunately, the solution by @Theo contains mistakes.\r
\n" ); document.write( "\n" ); document.write( "Some mistakes are technical (like typos).\r
\n" ); document.write( "\n" ); document.write( "But there are conceptual mistakes, too.\r
\n" ); document.write( "\n" ); document.write( "So, I came to put here the correct solution.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let x = # of girl dolls;\r\n" );
document.write( "    y = # of boy   dolls.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Your objective function is  R(x,y) = 9x + 9.5y which you want to minimize.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "your constraint functions are:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "12x + 14y <= 100000      (plastic constraint)      <<<---===  I edited this line\r\n" );
document.write( "5x        <= 30000       (silk constraint)\r\n" );
document.write( "x         >= 2y          (\"Kids will buy at least twice as many girl dolls as boys.\")     <<<---===  missed constrain\r\n" );
document.write( "x,y >= 0 (can't be negative)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Region of feasibility is shown in the plot below.\r\n" );
document.write( "\r\n" );
document.write( "It is the quadrilateral in Quadrant I below the blue line, below the red line and to the left from the vertical green line.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    //  To make the numbers on the coordinate axes better visible, I showed the values along coordinate axes in thousands dolls.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Plots y =  (red), x = 6 (green)  and    (blue)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The corner points of this region are where the maximum revenue will be.\r\n" );
document.write( "You will evaluate the objective function at these corner points to determine the corner point that provides the most revenue.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You also need to confirm that all constraints are satisfied as the corner point with the maximum revenue.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The corner point with the maximum revenue is at (6000,2000)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The revenue at this corner point was 73,000, which was higher than at any of the other corner points.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The plastic consumed was 100,000 and the nylon consumed was 30,000, all under the constraint limits of <= 100,000 and <= 30,000.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The cost function at the corner point was 9 * 6,000 + 9.5 * 2000 = 73,000\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The plastic constraint was 12 * 6000 + 14 * 2000 = 100,000\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The nylon constraint was 5 * 6000 + 0 * 2000 = 30,000\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );