document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=321938>Question 321938</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2.  The captain of a riverboat charges $36 per person including lunch.  The cruise averages 300 customers a day.  The captain is considering increasing the price.  A survey of the customers indicates that for every $2 increase, there would be 10 fewer customers.  What increase in price would maximize the revenue? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #230483 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=230483\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Cost is $36 per person including lunch.\r<BR>\n" );
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document.write( "Cruise averages 300 customers a day.\r<BR>\n" );
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document.write( "Current Revenue is modeled by  R = 36 * 300\r<BR>\n" );
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document.write( "Every $2 increase in price yields 10 fewer customers.\r<BR>\n" );
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document.write( "If you let x = change in price and change in number of customer, then you get an equation that looks like this:\r<BR>\n" );
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document.write( "R = (36 + 2x) * (300 - 10*x)\r<BR>\n" );
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document.write( "When x = 1, the price per person = 38 and the number of customers = 290.<BR>\n" );
document.write( "When x = 2, the price per person = 40 and the number of customers = 280.<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "By allowing x to represent a unit of change, you can now model what happens to the revenue based on a change in x.\r<BR>\n" );
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document.write( "The graph of this equation would look like this:\r<BR>\n" );
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document.write( "Formula to model this would be:\r<BR>\n" );
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document.write( "R = Total Revenue<BR>\n" );
document.write( "P = Price per person<BR>\n" );
document.write( "N = number of people.\r<BR>\n" );
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document.write( "Currently, P = 36 and N = 300 which makes R = 36 * 300 = 10800.\r<BR>\n" );
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document.write( "On the graph, that is the value of y when x = 0.\r<BR>\n" );
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document.write( "When x = 0, there is no change in the current price per person and the number of customers.\r<BR>\n" );
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document.write( "You can see from the graph that this curve will peak somewhere between x = 5 and x = 8.\r<BR>\n" );
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document.write( "This formula looks like a quadratic equation, so we can use the min/max formula of x = -b/2a to find the min/max point of this equation.\r<BR>\n" );
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document.write( "To do that, we need to transform this equation into the standard form of a quadratic equation which is:\r<BR>\n" );
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document.write( "ax^2 + bx + c = 0\r<BR>\n" );
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document.write( "Our equation is currently in the form of:\r<BR>\n" );
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document.write( "R = ((36 + 2x) * (300-10*x))\r<BR>\n" );
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document.write( "On the graph, R is represented by y, so we change the formula to show as:\r<BR>\n" );
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document.write( "y = ((36 + 2x) * (300-10*x))\r<BR>\n" );
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document.write( "We simplify this equation by using the distributive property of multiplication to get:\r<BR>\n" );
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document.write( "y  = 36 * (300 - 10*x) + 2x * (300 - 10*x)\r<BR>\n" );
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document.write( "we simplify further by performing the multiplications indicated to get:\r<BR>\n" );
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document.write( "y = 10800 - 360*x + 600*x - 20*x^2\r<BR>\n" );
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document.write( "We combine like terms to get:\r<BR>\n" );
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document.write( "y = 10800 + 240*x - 20*x^2\r<BR>\n" );
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document.write( "We set y = 0 to get:\r<BR>\n" );
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document.write( "0 = 10800 + 240*x - 20*x^2\r<BR>\n" );
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document.write( "We re-order the terms to get:\r<BR>\n" );
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document.write( "0 = -20*x^2 + 240*x + 10800\r<BR>\n" );
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document.write( "Since a = b implies that b = a, this equation can be rewritten as:\r<BR>\n" );
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document.write( "-20*x^2 + 240*x + 10800 = 0\r<BR>\n" );
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document.write( "The equation is now in standard form of a quadratic equation of:\r<BR>\n" );
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document.write( "ax^2 + bx + c and we get:\r<BR>\n" );
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document.write( "a = -20<BR>\n" );
document.write( "b = 240<BR>\n" );
document.write( "c = 10800\r<BR>\n" );
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document.write( "The x value of the max/min point of this quadratic equation is found by the formula of:\r<BR>\n" );
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document.write( "x = -b/2a which becomes:\r<BR>\n" );
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document.write( "x = -240 / -40 = 6\r<BR>\n" );
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document.write( "The x value of the max / min point is equal to 6.\r<BR>\n" );
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document.write( "The y value of the max / min point is equal to f(6).\r<BR>\n" );
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document.write( "Since y = f(x) = -20*x^2 + 240*x + 10800, then y = f(6) is equal to -20*(6)^2 + 240*(6) + 10800 which becomes:\r<BR>\n" );
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document.write( "y = -20*36 + 240*6 + 10800 which becomes:\r<BR>\n" );
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document.write( "y = -720 + 1440 + 10800 which becomes:\r<BR>\n" );
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document.write( "y = 11520\r<BR>\n" );
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document.write( "I'll repeat the graph with the addition of a horizontal line at y = 11520 so you can see that the maximum point of the graph is there.\r<BR>\n" );
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document.write( "The key was in setting up the equation on a 2 dimensional graph so that the value of y represented the revenue and the value of x represented both the increase in price and the drop in number of customers.\r<BR>\n" );
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document.write( "The only way I could figure to do this was to allow 2*x to equal the increase in price and 10*x equal the drop in numbers of customers.\r<BR>\n" );
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document.write( "y represents the value of the revenue.\r<BR>\n" );
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document.write( "x represents the value of a unit of change.\r<BR>\n" );
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document.write( "Once the formula was generated, the rest was fairly straightforward.\r<BR>\n" );
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document.write( "You just needed to know the standard form of a quadratic equation and the fact that the x value of the max / min point of a parabola (represented by a quadratic equation) was equal to -b/2a.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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