document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1147419>Question 1147419</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The revenue at the Assembly Center depends on the number of seats sold for the Willie Williams and the Wranglers concert. At $10 per ticket, they will fill all 8000 seats. The manager knows that for every $1 increase in the price, 500 tickets will go unsold. If the revenue in dollars, 𝑅(𝑝), is given by<BR>\n" );
document.write( "𝑅(𝑝) = −500𝑝^2 + 13000𝑝, where 𝑝 is the price per ticket sold.<BR>\n" );
document.write( "Find the number of unsold seats that resulted in this maximum revenue given the following 𝑛 = 13000 − 500𝑝 where 𝑛 is the number of seats sold and 𝑝 is the price.<BR>\n" );
document.write( "You must show this algebraically </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #768780 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=768780\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula given is r(p) = -500 * p^2 + 13000 * p<BR>\n" );
document.write( "at 10 dollars a ticket (p = 10), the revenue will be -500 * 10^2 + 13000 * 10<BR>\n" );
document.write( "that becomes -50,000 + 130,000 = 80,000.<BR>\n" );
document.write( "that, however, is not the maximum revenue.<BR>\n" );
document.write( "the equation given is a quadratic equation.<BR>\n" );
document.write( "to find the maximum revenue, set it equal to 0 and it will be in standard quadratic equation form.<BR>\n" );
document.write( "you will get:<BR>\n" );
document.write( "-500 * p^2 + 13000 * p = 0<BR>\n" );
document.write( "when in this form, a is equal to the coefficient of the p^2 term and b is equal to the coefficient of the p term, to get:<BR>\n" );
document.write( "a = -500<BR>\n" );
document.write( "b = 13000<BR>\n" );
document.write( "the maximum revenue will occur at p = -b / 2a = -13000 / -1000 = 13<BR>\n" );
document.write( "the maximum revenue will be when p = 13.<BR>\n" );
document.write( "the formula becomes r(p) = -500 * 13^2 + 13000 * 13 = 84,500.<BR>\n" );
document.write( "the number of seats will be 13 * 500 = 6,500.<BR>\n" );
document.write( "this is per the formula given of n = 13,000 - 500 * p.<BR>\n" );
document.write( "when p = 13, this formula becomes n = 13,000 - 500 * 13 = 13,000 - 6,500 = 6,500.<BR>\n" );
document.write( "since there are 8000 seats when the stadium is full, you will have 8000 - 6500 = 1500 empty seats when the revenue is maximized.<BR>\n" );
document.write( "you can graph the formula to see what's happening.<BR>\n" );
document.write( "the formula for graphing the revenue is y = -500x^2 + 13000x<BR>\n" );
document.write( "y replaces r(p) and x replaces p.<BR>\n" );
document.write( "the formula for graphing the number of seats sold is y = 13000 - 500 * x<BR>\n" );
document.write( "y replaces n and x replaces p.<BR>\n" );
document.write( "the graph looks like this.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/102201.jpg\" alt=\"$$$\" ><BR>\n" );
document.write( "the red line is the revenue and the blue line is the number of units sold.<BR>\n" );
document.write( "you can see that, when the price is 10, 8000 units are sold and the revenue is 80,000.<BR>\n" );
document.write( "you can also see that, when the price is 13, 6500 units are sold and the revenue is 84500.<BR>\n" );
document.write( "that's the maximum revenue and 1500 seats are empty because 8000 - 6500 = 1500.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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