document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202278>Question 1202278</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dances at the community centre produce revenue R=-60t^2 +600t, where R is the revenue and t the ticket price in dollars. Francine, the manager, found that the expenses, C, for the dances is modelled by C=162  -  120t. Note: Profit P = R- C.\r<BR>\n" );
document.write( "\n" );
document.write( "a) Determine the equation to represent the profit.<BR>\n" );
document.write( "b) Determine the break-even point (zero profit).<BR>\n" );
document.write( "c) Find the maximum profit and the ticket price that yields this profit. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836989 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=836989\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Revenue = money coming in<BR>\n" );
document.write( "Cost = money going out\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Profit = Revenue - Cost<BR>\n" );
document.write( "P = R - C<BR>\n" );
document.write( "P = ( R ) - ( C )<BR>\n" );
document.write( "P = ( -60t^2 +600t ) - ( 162 - 120t )<BR>\n" );
document.write( "P =  -60t^2 +600t  -  162 + 120t <BR>\n" );
document.write( "<font color=red>P =  -60t^2 + 720t - 162 </font> is the final answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "===================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The break-even point is when the company neither gains money nor loses money. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Set profit equal to zero to determine t.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P = 0<BR>\n" );
document.write( "-60t^2 + 720t - 162 = 0<BR>\n" );
document.write( "-6(10t^2 - 120t + 27) = 0<BR>\n" );
document.write( "10t^2 - 120t + 27 = 0/(-6)<BR>\n" );
document.write( "10t^2 - 120t + 27 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's use the quadratic formula.<BR>\n" );
document.write( "Plugging in a = 10, b = -120, c = 27<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"t+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-%28-120%29+%2B-+sqrt%28%28-120%29%5E2+-+4%2810%29%2827%29%29%29%2F%282%2810%29%29 ALIGN=MIDDLE  ALT=\"t+=+%28-%28-120%29+%2B-+sqrt%28%28-120%29%5E2+-+4%2810%29%2827%29%29%29%2F%282%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28120+%2B-+sqrt%2813320%29%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"t+=+%28120+%2B-+sqrt%2813320%29%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28120+%2B+sqrt%2813320%29%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"t+=+%28120+%2B+sqrt%2813320%29%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28120+-+sqrt%2813320%29%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"t+=+%28120+-+sqrt%2813320%29%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+11.77 ALIGN=MIDDLE  ALT=\"t+=+11.77\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+0.23 ALIGN=MIDDLE  ALT=\"t+=+0.23\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Each decimal value is approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font color=red>Answer:</font> <BR>\n" );
document.write( "Break-even point happens when t = 0.23 and when t = 11.77\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "===================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (c)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The vertex in this case is the highest point. It corresponds to the max profit.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The x coordinate of the vertex is the midpoint of the roots. <BR>\n" );
document.write( "Each root is a break-even point.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Average the two break-even points<BR>\n" );
document.write( "(0.23+11.77)/2 = 6<BR>\n" );
document.write( "The max profit happens when t = 6 is the ticket price.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug this into the profit function.<BR>\n" );
document.write( "P =  -60t^2 + 720t - 162<BR>\n" );
document.write( "P =  -60*6^2 + 720*6 - 162<BR>\n" );
document.write( "P =  1998\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font color=red>Answer:</font><BR>\n" );
document.write( "The max profit is $1,998. It occurs when the ticket price is $6.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );