document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209652>Question 1209652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> GetThere Airlines currently charges $200$ dollars per ticket and sells $40{,}000$ tickets a week. For every $10$ dollars they increase the ticket price, they sell $300$ fewer tickets a week. How many dollars should they charge to maximize their total revenue? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849740 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849740\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the ticket price that maximizes revenue for GetThere Airlines:\r<BR>\n" );
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document.write( "1. **Define variables:**\r<BR>\n" );
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document.write( "* Let 'x' be the number of $10 price increases.<BR>\n" );
document.write( "* Let 'P' be the ticket price.<BR>\n" );
document.write( "* Let 'Q' be the number of tickets sold.<BR>\n" );
document.write( "* Let 'R' be the total revenue.\r<BR>\n" );
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document.write( "2. **Express price and quantity in terms of x:**\r<BR>\n" );
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document.write( "* Price: P = 200 + 10x<BR>\n" );
document.write( "* Quantity: Q = 40000 - 300x\r<BR>\n" );
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document.write( "3. **Write the revenue function:**\r<BR>\n" );
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document.write( "Revenue is price times quantity:\r<BR>\n" );
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document.write( "R(x) = P * Q<BR>\n" );
document.write( "R(x) = (200 + 10x)(40000 - 300x)\r<BR>\n" );
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document.write( "4. **Expand the revenue function:**\r<BR>\n" );
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document.write( "R(x) = 8000000 - 60000x + 400000x - 3000x²<BR>\n" );
document.write( "R(x) = -3000x² + 340000x + 8000000\r<BR>\n" );
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document.write( "5. **Find the vertex of the parabola:**\r<BR>\n" );
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document.write( "Since the revenue function is a quadratic with a negative leading coefficient, it represents a parabola opening downwards. The vertex will give the maximum revenue. The x-coordinate of the vertex is given by:\r<BR>\n" );
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document.write( "x = -b / 2a<BR>\n" );
document.write( "x = -340000 / (2 * -3000)<BR>\n" );
document.write( "x = -340000 / -6000<BR>\n" );
document.write( "x = 56.67 (approximately)\r<BR>\n" );
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document.write( "6. **Determine the optimal price:**\r<BR>\n" );
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document.write( "Substitute the value of x back into the price equation:\r<BR>\n" );
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document.write( "P = 200 + 10x<BR>\n" );
document.write( "P = 200 + 10 * 56.67<BR>\n" );
document.write( "P = 200 + 566.7<BR>\n" );
document.write( "P = $766.70\r<BR>\n" );
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document.write( "7. **Round to the nearest dollar:**\r<BR>\n" );
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document.write( "Since we're dealing with money, we should round the price to the nearest dollar.  In this case, since x represents the number of $10 increases, we should use x = 56 or x = 57.\r<BR>\n" );
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document.write( "If x = 56, P = 200 + 10*56 = $760.<BR>\n" );
document.write( "If x = 57, P = 200 + 10*57 = $770.\r<BR>\n" );
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document.write( "Let's calculate the revenue for both:<BR>\n" );
document.write( "R(56) = (760)(40000 - 300*56) = 760 * 23200 = $17,632,000<BR>\n" );
document.write( "R(57) = (770)(40000 - 300*57) = 770 * 22900 = $17,633,000\r<BR>\n" );
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document.write( "So, the maximum revenue occurs at $770.\r<BR>\n" );
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document.write( "**Answer:** They should charge $770 per ticket to maximize their total revenue.<BR>\n" );
document.write( " </SPAN>\n" );
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