document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1065154>Question 1065154</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A small movie theater sells tickets for $15 at this price the theater sells 200 tickets every show the owners know in past years that they will sell eight more tickets per show for each price decrease of $0.50 what function can be used to model the owners earnings if x represents price decrease in dollars<BR>\n" );
document.write( "What lower price would let the owners earn the same amount of money they are now<BR>\n" );
document.write( "What should the owners charge per ticket to earn the maximum amount of money </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #680266 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52794)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=680266\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "A small movie theater sells tickets for $15 at this price the theater sells 200 tickets every show the owners know in past years <BR>\n" );
document.write( "that they will sell eight more tickets per show for each price decrease of $0.50 <BR>\n" );
document.write( "1. what function can be used to model the owners earnings if x represents price decrease in dollars<BR>\n" );
document.write( "2. What lower price would let the owners earn the same amount of money they are now<BR>\n" );
document.write( "3. What should the owners charge per ticket to earn the maximum amount of money<BR>\n" );
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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( "1.  N(x) = 200 + 16x, where x is the price decrease in dollars, N(x) is the number of viewers.\r\n" );
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document.write( "2.  To answer the second question, solve the equation \r\n" );
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document.write( "    N(x)*(15-x) = 15*200,   or, which is the same \r\n" );
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document.write( "   (200+16x)*(15-x) = 3000.\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+330%2C+-1.5%2C+20.5%2C+-200.5%2C+4000.5%2C%0D%0A++++++++++%28200%2B16x%29%2A%2815-x%29%2C+3000%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "Plots y = (200+16x)*(15-x) and y = 3000 \r\n" );
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document.write( "From the plot, you can see that the second solution is the price decrease of $2.50, which gives the ticket price of $12.50, \r\n" );
document.write( "the number of viewers of 240 and the total amount of money the same $15*200 = $3000.\r\n" );
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document.write( "3. To answer the third question, you need to find the maximum of the quadratic function\r\n" );
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document.write( "   q(x) = (200+16x)*(15-x) = -16x^2 + 40x + 3000.\r\n" );
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document.write( "   The maximum is achieved at x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-40%2F%282%2A%28-16%29%29 ALIGN=MIDDLE  ALT=\"-40%2F%282%2A%28-16%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%2F32 ALIGN=MIDDLE  ALT=\"40%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" > = $1.25.\r\n" );
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document.write( "   So, the optimal price decrease is $1.25. \r\n" );
document.write( "       The optimal price for a ticket is $13.75\r\n" );
document.write( "       It provides 220 viewers and the total money amount of 220*$13.75 = $3025.00.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "To be fully educated in solution of similar problems, see my other lessons in this site on finding the maximum/minimum of a quadratic function\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/Using-quadratic-functions-to-solve-problems-on-maximizing-profit.lesson>Using quadratic functions to solve problems on maximizing revenue/profit</A> (*)\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/OVERVIEW-of-lessons-on-finding-the-maximum-minimum-of-a-quadratic-function.lesson>OVERVIEW of lessons on finding the maximum/minimum of a quadratic function</A>\r<BR>\n" );
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document.write( "Pay a special attention to the lesson marked by (*) in the list.<BR>\n" );
document.write( "It contains other similar problems with detailed solutions.\r<BR>\n" );
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document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lessons are the part of this textbook under the topic \"<U>Finding minimum/maximum of quadratic functions</U>\". \r<BR>\n" );
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document.write( " </SPAN>\n" );
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