document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207808>Question 1207808</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sophia is organizing a movie night and wants to provide popcorn and soda to her guests. A bag of popcorn costs her $2 to buy, and a can of soda costs her $1. She only has room to store 24 cans of soda and 50 bags of popcorn at her house. She needs to have at least 60 items total to satisfy her guests. What is the least amount of money she can spend on popcorn and soda? ( This is a graphing question so if you read this do you think you could email me and I will send you the pictures of the questions o have 3 questions and if it’s necessary I am willing to pay for it.) Thank you i need the constraints,objetive function, Vertices of possible max./min, solutions, and of course the solution and graph ! Thank you so much  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845918 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=845918\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = number of bags of popcorn<BR>\n" );
document.write( "y = number of cans of soda\r<BR>\n" );
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document.write( "Since each bag of popcorn costs $2, she spends 2x dollars on popcorn alone.<BR>\n" );
document.write( "Each can of soda costs $1, meaning she spends 1y = y dollars on soda alone.<BR>\n" );
document.write( "2x+y = total amount spent for both items combined<BR>\n" );
document.write( "This will set up the objective function C = 2x+y.<BR>\n" );
document.write( "The goal is to make the output of this function the smallest possible, i.e. we want the smallest possible cost.\r<BR>\n" );
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document.write( "\"<font color=blue>She only has room to store 24 cans of soda and 50 bags of popcorn at her house</font>\" gives us these constraints <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C=+50 ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C=+50\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+y+%3C=+24 ALIGN=MIDDLE  ALT=\"0+%3C=+y+%3C=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "\"<font color=blue>She needs to have at least 60 items total to satisfy her guests</font>\" gives this constraint <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By+%3E=+60 ALIGN=MIDDLE  ALT=\"x%2By+%3E=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here are all of the constraints listed as a system of inequalities<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%280+%3C=+x+%3C=+50%2C0+%3C=+y+%3C=+24%2Cx%2By+%3E=+60%29 ALIGN=MIDDLE  ALT=\"system%280+%3C=+x+%3C=+50%2C0+%3C=+y+%3C=+24%2Cx%2By+%3E=+60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The first two inequalities define a rectangle that is 50 units across and 24 units tall. <BR>\n" );
document.write( "The lower left corner of this rectangle is at the origin (0,0). <BR>\n" );
document.write( "The upper right corner is at (50,24).\r<BR>\n" );
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document.write( "We'll overlap this rectangle with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By+%3E=+60 ALIGN=MIDDLE  ALT=\"x%2By+%3E=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the region where we shade above the line x+y = 60. <BR>\n" );
document.write( "Two points on this line are (0,60) and (60,0)\r<BR>\n" );
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document.write( "Here's what the final region looks like<BR>\n" );
document.write( "Any point in the purple region, or on the boundary, will satisfy all three constraint inequalities.<BR>\n" );
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document.write( "Graph created with Desmos.\r<BR>\n" );
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document.write( "Desmos link here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/9svzgkteyb\">https://www.desmos.com/calculator/9svzgkteyb</a>\r<BR>\n" );
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document.write( "The triangular region has the vertices:<BR>\n" );
document.write( "(50,10)<BR>\n" );
document.write( "(50,24)<BR>\n" );
document.write( "(36,24)\r<BR>\n" );
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document.write( "These vertices are determined by intersecting the various boundary lines. <BR>\n" );
document.write( "For example, solve the system {x+y=60,y=24} to determine the vertex location (36,24).\r<BR>\n" );
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document.write( "The idea is to plug those coordinates into the objective function to figure out which x,y pair gives the smallest value of C.<BR>\n" );
document.write( "Let's try the vertex (50,10)<BR>\n" );
document.write( "C = 2x+y<BR>\n" );
document.write( "C = 2*50+10<BR>\n" );
document.write( "C = 110\r<BR>\n" );
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document.write( "Now try the vertex (50,24)<BR>\n" );
document.write( "C = 2x+y<BR>\n" );
document.write( "C = 2*50+24<BR>\n" );
document.write( "C = 124<BR>\n" );
document.write( "So far the first vertex leads to the lowest cost.\r<BR>\n" );
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document.write( "Now the last vertex<BR>\n" );
document.write( "C = 2x+y<BR>\n" );
document.write( "C = 2*36+24<BR>\n" );
document.write( "C = 96<BR>\n" );
document.write( "This is the smallest cost of the trio.\r<BR>\n" );
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document.write( "Therefore, the lowest cost ($96) occurs when Sophia buys x = 36 bags of popcorn and y = 24 cans of soda.<BR>\n" );
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