document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173213>Question 1173213</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The event manager collected a total amount of not more than $40,000 from more than 200 adults and children who watched a basketball game. Tickets for that game cost<BR>\n" );
document.write( "$150 for adults and $100 for children. Around how many adults and children watched the game? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #798399 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=798399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your constaints are:<BR>\n" );
document.write( "x + y > 200<BR>\n" );
document.write( "150x + 100y <= 40000<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using the desmos.com calculator, you would graph the opposite of these inequalities.<BR>\n" );
document.write( "any point in the unshaded area of the graph and on the line line of 150x + 100y = 40000 that borders the unshaded area of the graph is valid.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3 valid points are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3 points that are valid are (0,400), (266.667,0) and (60,260).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "these points have to meet the requirements of the constraint inequalities.<BR>\n" );
document.write( "they're all greater than or equal to 0, so those requirements are met.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x + y must be > 200.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that requirement is met at all of the points mentioned since x + y is greater than 200 at all of those points.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "150x + 100y <= 40,000 has to be met at all of the points mentioned.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "150x + 100y <= 40,000 is evaluated at each of those points.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(0,400) = 100 * 400 = 40,000 which meets the requirements.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "266.667 * 150 = (266 + 2/3) * 150 = 800/3 * 150 <= 40,000 becomes 40,000 <= 40,000 which meets the requirements.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(60,260) = 150 * 60 + 150 * 260 = 35,000 <= 40,000, so that meets the requirements.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "all the points meet the requirement, so all of the points are valid.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the point (266.667,0), however does not lead to a whole number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that number should be rounded down or up to make it a whole number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(266,0) = 266 * 150 = 39900 <= 40,000, so rounding down is good.<BR>\n" );
document.write( "(267,0) = 267 * 150 = 40,050 which is not <= 40,0009, so rounding up is not good.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the 3 valid points that are whole numbers that meet the requirements are (0,400), (266,0), (60,260).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "around how many adults and children watched the game?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that number should be a minimum of 201 to a maximum of 400.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "both numbers will get you x + y > 200<BR>\n" );
document.write( "both numbers will get you 150x + 100y <= 40,000\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "maximum revenue with 0 adults and 400 children attending is 400 * 100 = 40,000.<BR>\n" );
document.write( "this meets the requirements.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "maximum revenue with 266 adults and 0 children attending is 266 * 150 = 39,900.<BR>\n" );
document.write( "this meets the requirements.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "unless i'm missing something, i don't think you can answer this equation with one number.<BR>\n" );
document.write( "there is a range of numbers.<BR>\n" );
document.write( "those numbers have to be in the unshaded portion of the graph as described above and shown below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's my opinion.<BR>\n" );
document.write( "here's the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/011801.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );