document.write( "Question 1142297: One gram of soybean meal provides at least 2.5 units of vitamins and 5 calories. One gram of meat byproducts provides at least 4.5 units of vitamins and 3 calories. One gram of grain provides at least 5 units of vitamins and 10 calories. If a gram of soybean meal costs 6 cents, a gram of meat byproducts 7 cents, and a gram of grain 9 cents, what mixture of these three ingredients will provide at least 48 units of vitamins and 54 calories per serving at minimum cost? What will be the minimum cost? \n" ); document.write( "
Algebra.Com's Answer #763028 by ikleyn(52813)![]() ![]() You can put this solution on YOUR website! . \n" ); document.write( " \r\n" ); document.write( "\r\n" ); document.write( "Let X = the mass of soybean meal consumed (in grams);\r\n" ); document.write( "\r\n" ); document.write( " Y = the mass of meat;\r\n" ); document.write( "\r\n" ); document.write( " Z = the mass of grain.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "The objective function to minimize is the cost\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( " C(X,Y,Z) = 6*X + 7*Y + 9*Z cents. (1)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "The constraints are\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( " 2.5*X + 4.5*Y + 5*Z >= 48 units of vitamins, (2)\r\n" ); document.write( "\r\n" ); document.write( " \r\n" ); document.write( " 5*X + 3*Y + 10*Z >= 54 calories. (3)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Other constraints are X >= 0; Y>= 0, and Z >= 0. (4)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Now, a remarkable fact is that the solution to this minimax problem can be obtained ANALYTICALLY.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Constraints (2) and (3) represent two planes in 3D:\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( " 2.5*X + 4.5*Y + 5*Z = 48 (5)\r\n" ); document.write( "\r\n" ); document.write( " 5*X + 3*Y + 10*Z = 54 (6)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "These planes are not parallel -- hence, their intersection is a straight line.\r\n" ); document.write( "\r\n" ); document.write( "The idea is to present this straight line in a parametric form - then the solution of the minimax problem\r\n" ); document.write( "on this straight line will be easy.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Multiply equation (5) by 2 (both sides) and then subtract equation (6) from the obtained equation.\r\n" ); document.write( "You will get\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( " 3Y - 9Y = 54 - 2*48, or -6Y = -42. \r\n" ); document.write( "\r\n" ); document.write( "Hence, \r\n" ); document.write( "\r\n" ); document.write( " Y = 7. (7)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Thus we found that the intersection of two planes (5) and (6) is a straight line, which lies on the plane Y = 7.\r\n" ); document.write( "\r\n" ); document.write( "Subctitute Y =7 into equations (5) and (6). You will get then\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( " 2.5*X + 4.5*7 + 5Z = 48 (5')\r\n" ); document.write( "\r\n" ); document.write( " 5*X + 3*7 + 10Z = 54, (6')\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "or, collecting all constant terms on the right side\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( " 2.5*X + 5Z = 16.5, (5'')\r\n" ); document.write( "\r\n" ); document.write( " 5*X + 10*Z = 33, (6'')\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Equations (5'') and (6'') are DEPENDENT (which is OBVIOUS).\r\n" ); document.write( "\r\n" ); document.write( "Hence, two equations (5'') and (6'') represent THE SAME plane.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "So, our straight line is the intersection of planes (7) and (6'').\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Now, from equation (6''),\r\n" ); document.write( "\r\n" ); document.write( " X = 6.6 - 2Z.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Thus our stright line in parametric form is\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( " X = 6.6 - 2Z, Y = 7. (8)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Substitute (8) into the objective function (1). You will get\r\n" ); document.write( "\r\n" ); document.write( " C(X,Y,Z) = 6*X + 7*Y + 9*Z = 6*(6.6 - 2Z) + 7*7 + 9*Z = 39.6 - 12Z + 49 + 9Z = -3Z + 88.6. (9)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Thus, on our line the objective function is presented as the linear function (9) of Z.\r\n" ); document.write( "\r\n" ); document.write( "We see that when Z increases from 0 to positive values, the function (9) decreases.\r\n" ); document.write( "But Z can increase only till X = 6.6 - 2Z is >= 0 (is non-negative).\r\n" ); document.write( "Hence, the linear function (9) has the minimum at Z = 6.6/2 = 3.3.\r\n" ); document.write( "\r\n" ); document.write( "Then X = 6.6 - 2*3.3 = 0, according to (8).\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Thus we just obtained the solution to our minimax problem:\r\n" ); document.write( "\r\n" ); document.write( " The minimum solution point is X= 0; Y= 7 and Z= 3.3 \r\n" ); document.write( "\r\n" ); document.write( " and the mimimum cost is -3*Z + 88.6 = -3*3.3 + 88.6 = 78.7.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "ANSWER. The minimum cost is 78.70 cents and it is achieved at this diet:\r\n" ); document.write( "\r\n" ); document.write( " 0 gram of soybean meal; 7 gram of meat, and 3.3 gram of grain.\r\n" ); document.write( "\r \n" ); document.write( "\n" ); document.write( "Solved.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |