document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196212>Question 1196212</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rent cost for a large bus is #800 and #600 for the small bus. Calculate how many buses of each type should be used for the trip for the least possible cost. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828952 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828952\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let <BR>\n" );
document.write( "A = larger bus  (50 seats)<BR>\n" );
document.write( "B = smaller bus (40 seats)\r<BR>\n" );
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document.write( "Bus A costs $800<BR>\n" );
document.write( "Bus B costs $600\r<BR>\n" );
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document.write( "x = number of buses of type A<BR>\n" );
document.write( "y = number of buses of type B\r<BR>\n" );
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document.write( "Bus A can hold 50 students, so x of them contribute 50x seats<BR>\n" );
document.write( "Bus B can hold 40 students, so y of them contribute 40y seats<BR>\n" );
document.write( "In total there are 50x+40y seats available\r<BR>\n" );
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document.write( "We want that seat count to be 400 or larger. <BR>\n" );
document.write( "We may or may not have extra/empty seats.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=total_seats_available+%3E=+400 ALIGN=MIDDLE  ALT=\"total_seats_available+%3E=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50x%2B40y+%3E=+400 ALIGN=MIDDLE  ALT=\"50x%2B40y+%3E=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%285x%2B4y%29+%3E=+400 ALIGN=MIDDLE  ALT=\"10%285x%2B4y%29+%3E=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B4y+%3E=+400%2F10 ALIGN=MIDDLE  ALT=\"5x%2B4y+%3E=+400%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B4y+%3E=+40 ALIGN=MIDDLE  ALT=\"5x%2B4y+%3E=+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is one of the restrictions.\r<BR>\n" );
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document.write( "Another restriction is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By+%3C=+9 ALIGN=MIDDLE  ALT=\"x%2By+%3C=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > since x+y is the total number of buses used (of both types combined). <BR>\n" );
document.write( "We only have 9 drivers, so this total must be 9 or fewer.\r<BR>\n" );
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document.write( "Now either graph by hand, or use a graphing app such as Desmos or GeoGebra. <BR>\n" );
document.write( "In the screenshot below, I'm using GeoGebra<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/56Il4Ns.png\"><BR>\n" );
document.write( "The shaded triangle consists of all possible solutions to make both inequalities true, and also to keep x > 0 and y > 0<BR>\n" );
document.write( "It makes no sense to have x or y be negative.<BR>\n" );
document.write( "Take note that the green shaded region is below the boundary x+y = 9, and above the boundary 5x+4y = 40, and also above the x axis.\r<BR>\n" );
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document.write( "When it comes to finding the min cost, we'll be looking at the vertices. This is standard for linear programming problems.\r<BR>\n" );
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document.write( "The shaded triangle has vertices of:<BR>\n" );
document.write( "(4,5)<BR>\n" );
document.write( "(8,0)<BR>\n" );
document.write( "(9,0)\r<BR>\n" );
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document.write( "Plug each of those x,y coordinates into the cost function<BR>\n" );
document.write( "C(x,y) = 800x + 600y\r<BR>\n" );
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document.write( "Plug in (x,y) = (4,5)<BR>\n" );
document.write( "C(x,y) = 800x + 600y<BR>\n" );
document.write( "C(4,5) = 800*4 + 600*5<BR>\n" );
document.write( "C(4,5) = 3200 + 3000<BR>\n" );
document.write( "C(4,5) = 6200\r<BR>\n" );
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document.write( "Repeat for (x,y) = (8,0)<BR>\n" );
document.write( "C(x,y) = 800x + 600y<BR>\n" );
document.write( "C(8,0) = 800*8 + 600*0<BR>\n" );
document.write( "C(8,0) = 6400 + 0<BR>\n" );
document.write( "C(8,0) = 6400\r<BR>\n" );
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document.write( "And lastly do so for (x,y) = (9,0)<BR>\n" );
document.write( "C(x,y) = 800x + 600y<BR>\n" );
document.write( "C(9,0) = 800*9 + 600*0<BR>\n" );
document.write( "C(9,0) = 7200 + 0<BR>\n" );
document.write( "C(9,0) = 7200\r<BR>\n" );
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document.write( "The cost outputs were: 6200, 6400, and 7200<BR>\n" );
document.write( "The smallest of which is 6200 and this corresponds to when (x,y) = (4,5)\r<BR>\n" );
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document.write( "Answer: They should use 4 buses of type A (the 50 seater buses) and 5 buses of type B (the 40 seater buses) to achieve the lowest cost of $6200<BR>\n" );
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