document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=550973>Question 550973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A town is organizing a parade. There will be 2 float sizes. One is 30ft and the other is 15ft.  A 10 ft space will be left after each float. \r<BR>\n" );
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document.write( "A.The parade must be a least 150ft long, but less than 200ft long. What combinations of large and small floats are possible?\r<BR>\n" );
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document.write( "B. Large float costs $600 to operate. Small floats cost $300 to operate. The town has a budget of $2500 to operate the floats. How does this change your answers to part A? What combinations of large and small floats are possible? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #359305 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=359305\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your cost equation is:<BR>\n" );
document.write( "c = 600x + 300y<BR>\n" );
document.write( "your cost has to be less than or equal to 2500, so your cost constraint becomes:<BR>\n" );
document.write( "600x + 300y <= 2500<BR>\n" );
document.write( "anything that costs over 2500 is rejected.<BR>\n" );
document.write( "x = number of large floats<BR>\n" );
document.write( "y = number of small floats<BR>\n" );
document.write( "the formula for the maximum number of large floats is:<BR>\n" );
document.write( "40x - 10 <= 200<BR>\n" );
document.write( "this becomes 40x <= 210 which becomes:<BR>\n" );
document.write( "x <= 210/40<BR>\n" );
document.write( "this means that x has to be <= 5.25 which means that x has to be <= 5 since x has to be an integer.<BR>\n" );
document.write( "the formula for the maximum number of small floats is:<BR>\n" );
document.write( "25y - 10 <= 200<BR>\n" );
document.write( "this becomes 25x <= 210 which becomes:<BR>\n" );
document.write( "y <= 210/25<BR>\n" );
document.write( "this means that y has to be <= 8.4 which means that y has to be <= 8 since y has to be an integer.<BR>\n" );
document.write( "so far the constraints are:<BR>\n" );
document.write( "x <= 5<BR>\n" );
document.write( "y <= 8<BR>\n" );
document.write( "the formula for the minimum number of floats in the parade is:<BR>\n" );
document.write( "40x + 25y - 10 >= 150<BR>\n" );
document.write( "add 10 to each side of this equation to get:<BR>\n" );
document.write( "40x + 25y >= 160<BR>\n" );
document.write( "the formula for the maximum number of floats in the parade is:<BR>\n" );
document.write( "40x + 25y - 10 <= 200<BR>\n" );
document.write( "add 10 to each side of this equation to get:<BR>\n" );
document.write( "40x + 25y <= 210<BR>\n" );
document.write( "the cost equation for this problem is:<BR>\n" );
document.write( "600x + 300y <= 2500<BR>\n" );
document.write( "this equation is both a constraint and the objective function of this problem.<BR>\n" );
document.write( "we want the minimum cost and the cost has to be less than or equal to 2500.<BR>\n" );
document.write( "we solve this equation graphically by doing the following:<BR>\n" );
document.write( "graph the equation of y = 8<BR>\n" );
document.write( "graph the equation of x = 5<BR>\n" );
document.write( "graph the equation of 40x + 25y <= 210<BR>\n" );
document.write( "graph the equation of 40x + 25y >= 160<BR>\n" );
document.write( "graph the equation of 600x + 300y <= 2500<BR>\n" );
document.write( "your graph will look something like the following:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-3%2C7%2C-1%2C10%2C8%2C1000%28x-5%29%2C%28160-40x%29%2F25%2C%28210-40x%29%2F25%2C%282500-600x%29%2F300%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "your area of compatibility will be below the graph of 600x + 300y = 2500 and above the graph of 40x + 25y = 160.<BR>\n" );
document.write( "that area is shown in the following diagram.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2011/dec302.jpg\" alt = \"$$$$$\" /><BR>\n" );
document.write( "it is within this region that all constraints are met.<BR>\n" );
document.write( "all combinations within that region will have a parade length greater than or equal to 150 feet and less than or equal to 200 feet and will come within the maximum cost of 2500.<BR>\n" );
document.write( "based on the properties of linear programming, the minimum / maximum points of the objective function are at the corners of the compatible region.<BR>\n" );
document.write( "in this graph, those corners would be (x,y) coordinate points of:<BR>\n" );
document.write( "(4,0)<BR>\n" );
document.write( "(0,7)<BR>\n" );
document.write( "(0,8)<BR>\n" );
document.write( "while (7,0) and (8,0) are technically not at the corners, they are the nearest integers to those corners and one of the requirements are that x and y are integers.<BR>\n" );
document.write( "when x = 4, the number of floats in the parade are 4 large and 0 small and the total cost is 600 * 4 = 2400 which is under the limit and the total number of feet required is 4*40 - 10 = 150 which is within the limits of 150 t 200 feet.<BR>\n" );
document.write( "when y = 7, the number of floats in the parade are 0 large and 7 small and the total cost is 300 * 7 = 2100 which is under the limit and the total number of feet required is 7*25 - 10 = 165 which is within the limits of 150 to 200 feet.<BR>\n" );
document.write( "when y = 8, the number of floats in the parade are 0 large and 8 small and the total cost is 300 * 8 = 2400 which is under the limit and the total number of feet required is 8* 25 - 10 = 190 feet which is within the limits of 150 to 200 feet.<BR>\n" );
document.write( "the minimum cost combination is 0 large floats and 7 small floats.<BR>\n" );
document.write( "there are other possible combinations that will be within the compatibility region.<BR>\n" );
document.write( "those possible combinations are shown in the following table:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "large floats       small floats      feet required      cost\r\n" );
document.write( "0                      7                  165            2100\r\n" );
document.write( "0                      8                  190            2400\r\n" );
document.write( "1                      5                  155            2100\r\n" );
document.write( "1                      6                  180            2400\r\n" );
document.write( "2                      4                  170            2400\r\n" );
document.write( "3                      2                  160            2400\r\n" );
document.write( "4                      0                  150            2400\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "the minimum cost is at (0,7).<BR>\n" );
document.write( "it is also at (1,5).<BR>\n" );
document.write( "this stands to reason since 1*600 = 2*300 so you can exchange 1 large float for 2 small floats from a cost perspective and break even.<BR>\n" );
document.write( "the fact that (1,5) is also a minimal cost does not negate the fact that the minimum or maximum value will be at the corners.<BR>\n" );
document.write( "it still is, even though a value not at a corner is also a minimal cost.<BR>\n" );
document.write( " </SPAN>\n" );
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