document.write( "Question 1148051: Use the graphical method to solve the following question. Scale: 2 cm: 10 units on both axes.
\n" ); document.write( "A butcher produces sausage from a blend of beef, pork and fat. Batches of 50 kg of the mixture are made up. He must ensure that there is at most one quarter the amount of pork as there is beef. The fat is used as a filler and each 50 kg portion of sausage should not contain more than 10 kg of the latter to maintain the quality. The costs per kg of beef, pork and fat are N$6-00, N$4-00 and N$2-00 respectively. Determine how the 50 kg batches should be made up to minimize the costs and calculate the minimum cost.
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #769495 by ikleyn(52788)\"\" \"About 
You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            This problem can be solved  VERY  QUICKLY,  by the very simple way and without using the Linear programming approach.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "Let  x = the amount of beef, in kilograms;\r\n" );
document.write( "     y = the amount of pork;\r\n" );
document.write( "     z = the amount of fat.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Your constraint equations are:\r\n" );
document.write( "\r\n" );
document.write( "     x + y + z = 50\r\n" );
document.write( "     z <= 10\r\n" );
document.write( "     y <= 0.25x\r\n" );
document.write( "     x >= 0, y >= 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Your objective function to minimize is\r\n" );
document.write( "\r\n" );
document.write( "     cost = 6x + 4y + 2z\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Since fat is cheapest ingredient, the butcher decides to use all 10 kg of fat.\r\n" );
document.write( "Than your constraints take the form\r\n" );
document.write( "\r\n" );
document.write( "     x + y = 40          (1)\r\n" );
document.write( "     y <= 0.25x          (2)\r\n" );
document.write( "     x >= 0, y >= 0      (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and your task now is to minimize the form\r\n" );
document.write( "\r\n" );
document.write( "     cost = 6x + 4y + 20\r\n" );
document.write( "\r\n" );
document.write( "which is the same as to minimize\r\n" );
document.write( "\r\n" );
document.write( "     cost' = 6x + 4y.    (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, your problem is just reduced to 2 variables.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, from (1), express  y = 40-x  and substitute it into (4) and (2). You will get the function to minimize\r\n" );
document.write( "\r\n" );
document.write( "     cost' = 6x + 4*(40-x) = 160 + 2x    (5)\r\n" );
document.write( "\r\n" );
document.write( "under the restriction (2)\r\n" );
document.write( "\r\n" );
document.write( "    40-x <= 0.25x,  or  40 <= x + 0.25x = 1.25x, which is  x >= \"40%2F1.25\" = 32.   (6)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Formula (5) says \"than lesser is x, than lesser is the value of the function c' = 160+2x.\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "But formula (6) says that you CAN NOT take x lesser than 32.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus,  x= 32 kilograms of beef is the solution.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the full and complete ANSWER is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    x= 32 kilograms of beef,  y= 40-32 = 8 kilograms of pork and z= 10 kilograms of fat.\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved (!)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved even without using Linear programming approach  (!)  (!)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved at the level accessible even for  6 - 7- 8  grade middle school students level  (!)  (!!)  (!!!)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );