document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187890>Question 1187890</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A small firm manufactures and sells one liter cartons of non-alcoholic cocktails, “The Caribbean” and “Mr. Fruity”, which sell for $1.00 and $1.25, respectively. Each is made by mixing fresh orange, pineapple, and apple juices in different proportions. “The Caribbean” consists of 1 part orange, 6 parts pineapple, and 1 part apple. “Mr. Fruity” consists of 2 parts orange, 3 parts pineapple, and 1 part apple. The firm can buy up to 300 L of orange juice, up to 1125 L of pineapple juice, and up to 195 L of apple juice each week at a cost of $0.74, $0.64, and $0.48 per liter, respectively. Find the number of cartons of the two drinks that the firm should produce in order to maximize profit. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #818944 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=818944\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = the number of liters of the caribbean.<BR>\n" );
document.write( "let y = the number of liters of mister fruity.\r<BR>\n" );
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document.write( "one carton of either drink is equal to 1 liter.\r<BR>\n" );
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document.write( "the caribbean consists of 1 part orange juice, 6 parts pineapple juice, and 1 part apple juice.<BR>\n" );
document.write( "mister fruity consists of 2 parts orange juice, 3 parts pineapple juice, and 1 part apple juice.\r<BR>\n" );
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document.write( "total number of parts for the caribbean is 8.<BR>\n" );
document.write( "total number of parts for mister fruity is 6.\r<BR>\n" );
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document.write( "since the total number of liters of each drink is 1, then:\r<BR>\n" );
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document.write( "each part of the caribbean is 1/8 of 1 liter.<BR>\n" );
document.write( "each part of mister fruity is 1/6 of 1 liter.\r<BR>\n" );
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document.write( "since the caribbean consists of 1 part orange juice, 6 parts pineapple juice, and 1 part apple juice, then the caribbean uses 1/8 of a liter of orange juice, 6/8 of a liter of pineapple juice, 1/8 of a liter of apple juice.\r<BR>\n" );
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document.write( "since mister fruity consists of 2 parts orange juice, 3 parts pineapple juice, and 1 part apple juice, then mister fruity uses 2/6 of a liter of orange juice, 3/6 of a liter of pineapple juice, 1/6 of a liter of apple juice.\r<BR>\n" );
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document.write( "the firm can buy up to: 300 liters of orange juice, 1125 liters of pineapple juice, 195 liters of apple juice.\r<BR>\n" );
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document.write( "since the caribbean uses 1/8 of a liter of orange juice and mister fruity uses 2/6 of a liter of orange juice, then your constraint equation for orange juice is:\r<BR>\n" );
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document.write( "1/8 * x + 2/6 * y <= 300\r<BR>\n" );
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document.write( "since the caribbean uses 6/8 of a liter of pineapple juice and mister fruity uses 3/6 of a liter of pineapple juice, then your constraint equation for pineapple is:\r<BR>\n" );
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document.write( "6/8 * x + 3/6 * y <= 1125\r<BR>\n" );
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document.write( "since the caribbean uses 1/8 of a liter of apple juice and mister fruity uses 1/6 of a liter of apple juice, then your constraint equation for apple juice is:\r<BR>\n" );
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document.write( "1/8 * x + 1/6 * y <= 195\r<BR>\n" );
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document.write( "another constraint is that both x and y must be greater than or equal to 0.\r<BR>\n" );
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document.write( "the cost for orange juice is .74 per liter.<BR>\n" );
document.write( "the cost for pineapple juice is .64 per liter.<BR>\n" );
document.write( "the cost for apple juice is .48 per liter.\r<BR>\n" );
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document.write( "the selling price of the caribbean is 1 dollar per liter.<BR>\n" );
document.write( "the selling price of mister fruity is 1.25 dollars per liter.\r<BR>\n" );
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document.write( "the profit equation is the selling price minus the cost.\r<BR>\n" );
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document.write( "for the caribbean, the profit equation will be:<BR>\n" );
document.write( "1.00 - 1/8 * .74 - 6/8 * .64 - 1/8 * .48 = .3675.\r<BR>\n" );
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document.write( "for mister fruity, the profit equation will be:<BR>\n" );
document.write( "1.25 - 2/6 * .74 - 3/6 * .64 - 1/6 * .48 = .6033333333.\r<BR>\n" );
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document.write( "the profit equation is your objective function and is shown below:<BR>\n" );
document.write( "profit = .3675 * x + .6033333333 * y\r<BR>\n" );
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document.write( "the constraint functions are:<BR>\n" );
document.write( "1/8 * x + 2/6 * y <= 300 (orange juice)<BR>\n" );
document.write( "6/8 * x + 3/6 * y <= 1125 (pineapple juice)<BR>\n" );
document.write( "1/8 * x + 1/6 * y <= 195 (apple juice)<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "using the desmos.com calculator, you will graph the opposite of the inequalities.<BR>\n" );
document.write( "the area of the graph that is not shaded is your region of feasibility.\r<BR>\n" );
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document.write( "the corner points of the region of feasibility will contain the maximum profit.\r<BR>\n" );
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document.write( "each of those corner points will be evaluated by the objective function to determine which of them contains the maximum profit.\r<BR>\n" );
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document.write( "the coordinate points are in (x,y) format.<BR>\n" );
document.write( "x represents the number of liters of the caribbean.<BR>\n" );
document.write( "y represents the number of liters of mister fruity.\r<BR>\n" );
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document.write( "the graph looks like this:\r<BR>\n" );
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document.write( "all the constraints must be satisfied.\r<BR>\n" );
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document.write( "x >= 0 and y >= 0 is satisfied.\r<BR>\n" );
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document.write( "1/8 * x + 2/6 * y <= 300 (orange juice) is satisfied because 1/8 * 720 + 2/6 * 630 = 300 which is equal to 300.\r<BR>\n" );
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document.write( "6/8 * x + 3/6 * y <= 1125 (pineapple juice) is satisfied because 720 * 6/8 + 3/6 * 630 = 855 which is less than 1125.\r<BR>\n" );
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document.write( "1/8 * x + 1/6 * y <= 195 (apple juice) is satisfied because 720 * 1/8 + 630 * 1/6 = 168.75 which is less than 195.\r<BR>\n" );
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document.write( "if i did this right (always a consideration), your solution is that 720 cartons of the carribean and 630 cartons of mister fruity should be produced and sold to generate the maqximum profit.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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