document.write( "Question 1068942: Christopher is making necklace and key chains. He can make a necklace in 0.5 hours and key chain in 0.25 hours and he has no more than 20 hours. The cost in making a necklace is $2 while for key chain it is $3.He could invest up-to $120.write a system of four linear inequalities andfind how many of each he could make? If he sells each necklace for $10 and keychain for $8, tell when he will get maximum revenue and maximum profit? \n" ); document.write( "
Algebra.Com's Answer #684182 by ikleyn(52794)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "Christopher is making necklace and key chains.
\n" ); document.write( "He can make a necklace in 0.5 hours and key chain in 0.25 hours and he has no more than 20 hours.
\n" ); document.write( "The cost in making a necklace is $2 while for key chain it is $3. He could invest up-to $120.
\n" ); document.write( "Write a system of four linear inequalities and find how many of each he could make?
\n" ); document.write( "If he sells each necklace for $10 and key chain for $8, tell when he will get maximum revenue and maximum profit?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "Let X be the number of necklaces and Y be the number of key chains.\r\n" );
document.write( "\r\n" );
document.write( "Then the inequalities are\r\n" );
document.write( "\r\n" );
document.write( "X >= 0;\r\n" );
document.write( "\r\n" );
document.write( "Y >= 0;\r\n" );
document.write( "\r\n" );
document.write( "0.5X + 0.25Y <= 20   (the restriction by time);\r\n" );
document.write( "\r\n" );
document.write( "2X + 3Y <= 120       (the restriction by investing)\r\n" );
document.write( "\r\n" );
document.write( "The inequalities describe and represent this quadrilateral in the first quadrant,\r\n" );
document.write( "restricted by the x- and y-axes and the red and the green lines:\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 = \"%2820-0.5x%29%2F0.25\" (red)  and y = \"%28120-2x%29%2F3\" (green)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The vertices of this quadrilateral are \r\n" );
document.write( "\r\n" );
document.write( "P1 = (0,0)     = the origin of the coordinate system,\r\n" );
document.write( "\r\n" );
document.write( "P2 = (0,60)    = the y-intersect of the green line,\r\n" );
document.write( "\r\n" );
document.write( "P3 = (40,0)    = the x-intersect of the red   line,  and\r\n" );
document.write( "\r\n" );
document.write( "P4 = (30,20)   = the intersection point of the red and the green lines.\r\n" );
document.write( "\r\n" );
document.write( "                 (this intersection point is the solution of the system  0.5X + 0.25Y = 20  and  2X + 3Y = 120).\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "The solution for the maximum profit\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "The profit function is P(X,Y) = ($10 - $2)*X + ($8 - $3)Y = 8X + 5Y dollars.    (1)\r\n" );
document.write( "\r\n" );
document.write( "To find the maximum profit, you should check the values of the profit function P(x,Y) at all four vertices and then compare four values.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The values are: \r\n" );
document.write( "\r\n" );
document.write( "at P1:  P(X,Y) = P(0,0)   = 8*0  + 5*0   =   0;\r\n" );
document.write( "\r\n" );
document.write( "at P2:  P(X,Y) = P(0,60)  = 8*0  + 5*60  = 300;\r\n" );
document.write( "\r\n" );
document.write( "at P3:  P(X,Y) = P(40,0)  = 8*40 + 5*0   = 320;\r\n" );
document.write( "\r\n" );
document.write( "at P4:  P(X,Y) = P(30,20) = 8*30 + 5*20  = 340.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The maximal value of the profit function is at P4, and it gives the optimal solution: \r\n" );
document.write( "\r\n" );
document.write( "      The maximum profit is $340 and it is achieved when Christoper produces 30 necklaces and 20 key chains.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "The solution for the maximum revenue\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "The revenue function is R(X,Y) = $10*X + $8*Y = 10X + 8Y dollars.               (2)\r\n" );
document.write( "\r\n" );
document.write( "To find the maximum revenue, you should check the values of the revenue function R(x,Y) (2) at all four vertices and then compare four values.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The values are: \r\n" );
document.write( "\r\n" );
document.write( "at P1:  R(X,Y) = R(0,0)   = 10*0  + 8*0   =   0;\r\n" );
document.write( "\r\n" );
document.write( "at P2:  R(X,Y) = R(0,60)  = 10*0  + 8*60  = 480;\r\n" );
document.write( "\r\n" );
document.write( "at P3:  R(X,Y) = R(40,0)  = 10*40 + 8*0   = 400;\r\n" );
document.write( "\r\n" );
document.write( "at P4:  R(X,Y) = R(30,20) = 10*30 + 8*20  = 460.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The maximal value of the revenue function is at P2, and it gives the optimal solution for the revenue: \r\n" );
document.write( "\r\n" );
document.write( "      The maximum revenue is $480 and it is achieved when Christoper produces 0 necklaces and 60 key chains.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "As you see, maximum profit and maximum revenue are achieved at different points and assume different strategies.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The method I solved this problem is called \"the linear programming method\".\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It may have different / (other) names, too.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );