document.write( "Question 727599: I have a graph that I am supposed to fill out along with answering the equation. The problem is this: \r
\n" ); document.write( "\n" ); document.write( "I need to have some stuffed animals for a kid’s party. I want to have 24 cats, 20 dogs and 12 dragons. (If I end up with extras, that is fine.) To save money, I will buy the big party assortments. There are two assortments available. The Stuffed-Animals-A-Plenty, for $5 has 3 cats, 2 dogs, and 4 dragons. The Kids-Box, for $6 has 4 cats, 4 dogs, and 3 dragons. How many of each party assortment should I buy to minimize my cost? \n" ); document.write( "
Algebra.Com's Answer #445146 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let x = the number of Stuffed-Animals-A-Plenty's, which means we have\r\n" );
document.write( "3x cats, 2x dogs, and 4x dragons, at a cost of $5x\r\n" );
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document.write( "Let y = the number of Kids-Box's, which means we have\r\n" );
document.write( "4y cats, 4y dogs, and 3y dragons, at a cost of $6y\r\n" );
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document.write( "3x + 4y ≧ 24     <---- the cat inequality\r\n" );
document.write( "2x + 4y ≧ 20     <---- the dog equality\r\n" );
document.write( "4x + 3y ≧ 12     <---- the dragon equality\r\n" );
document.write( "      x ≧ 0      <---- x cannot be negative\r\n" );
document.write( "      y ≧ 0      <---- y cannot be negative\r\n" );
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document.write( "The objective function (money equation) to minimize is\r\n" );
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document.write( "C(x,y) = 5x + 6y   \r\n" );
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document.write( "The equations of the lines are\r\n" );
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document.write( "3x + 4y = 24     <---- the boundary equation for the cats\r\n" );
document.write( "2x + 4y = 20     <---- the boundary equation for the dogs\r\n" );
document.write( "4x + 3y = 12     <---- the boundary equation for the dragons. \r\n" );
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document.write( "Using the origin as a test point shows that all solutions\r\n" );
document.write( "are above all three lines. The blue line can be eliminated\r\n" );
document.write( "since it is below both of the other lines. \r\n" );
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document.write( "Solve the system\r\n" );
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document.write( "3x + 4y = 24    \r\n" );
document.write( "2x + 4y = 20   \r\n" );
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document.write( "and get solution x=4, y=3 which is the point (4,3).  So\r\n" );
document.write( "the corner points are the y-intercept of the red line,\r\n" );
document.write( "which is (0,6) and the x-intercept of the green line, which is (10,0)\r\n" );
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document.write( "The feasible region consists of all the points on or above these lines.\r\n" );
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document.write( "We evaluate the objective function at the three corner points:\r\n" );
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document.write( "Corner | \r\n" );
document.write( "Point  |  C(x,y) = 5x + 6y\r\n" );
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document.write( " (0,6) |  C(0,6) = 5·0  + 6·6 =  0 + 36 = 46\r\n" );
document.write( " (4,3) |  C(4,3) = 5·4  + 6·3 = 20 + 18 = 38 \r\n" );
document.write( "(10,0) | C(10,0) = 5·10 + 6·0 = 50 +  0 = 50\r\n" );
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document.write( "Therefore the minimum cost is $38 when 4 Stuffed-Animals-A-Plenty's\r\n" );
document.write( "and 3 Kids-Boxes are purchased.\r\n" );
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document.write( "Edwin
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