document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172441>Question 1172441</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 800 fewer tickets a week. How many dollars should they charge to maximize their total revenue?\r<BR>\n" );
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document.write( "I have gotten the answer of $300 by doing these steps. What have I done wrong?\r<BR>\n" );
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document.write( "Number of tickets = 40000 - 1000 * (P - 200)/10\r<BR>\n" );
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document.write( "Revenue = (40000 - 1000 * (P - 200)/10) * P\r<BR>\n" );
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document.write( "= 60000 P - 100 P^2\r<BR>\n" );
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document.write( "60000 - 200 P = 0\r<BR>\n" );
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document.write( "P = 300\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
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document.write( "Currently the revenue is <BR>\n" );
document.write( "revenue = (number of tickets sold)*(price per ticket)<BR>\n" );
document.write( "revenue = (40,000)*(200)<BR>\n" );
document.write( "revenue = 8,000,000 dollars\r<BR>\n" );
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document.write( "Let x be the number of times they increase the ticket price by $10<BR>\n" );
document.write( "x is some nonnegative whole number.\r<BR>\n" );
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document.write( "The old price was 200 dollars<BR>\n" );
document.write( "The new price is 200+10x dollars<BR>\n" );
document.write( "Eg: if they increase it 3 times, then 200+10x=200+10*3 = 230 dollars is the new ticket price\r<BR>\n" );
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document.write( "Each time they increase the price by $10, the amount sold goes down by 800<BR>\n" );
document.write( "old amount sold = 40,000<BR>\n" );
document.write( "new amount sold = 40,000 - 800x<BR>\n" );
document.write( "Eg: if they increased it 3 times, then <BR>\n" );
document.write( "40000-800x = 40000-800*3 = 37,600 is the amount of tickets sold.\r<BR>\n" );
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document.write( "The new revenue function is therefore<BR>\n" );
document.write( "R(x) = (number of tickets sold)*(price per ticket)<BR>\n" );
document.write( "R(x) = (40000-800x)*(200+10x)<BR>\n" );
document.write( "R(x) = 40000*(200+10x)-800x*(200+10x)<BR>\n" );
document.write( "R(x) = 8000000+400000x-160000x-8000x^2<BR>\n" );
document.write( "R(x) = -8000x^2 + 240000x + 8000000\r<BR>\n" );
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document.write( "The simplified result is of the form<BR>\n" );
document.write( "y = ax^2 + bx + c<BR>\n" );
document.write( "where<BR>\n" );
document.write( "a = -8000<BR>\n" );
document.write( "b = 240000<BR>\n" );
document.write( "c = 8000000\r<BR>\n" );
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document.write( "Use the values of 'a' and b to find the x coordinate of the vertex h<BR>\n" );
document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -240000/(2*(-8000))<BR>\n" );
document.write( "h = -240000/(-16000)<BR>\n" );
document.write( "h = 15<BR>\n" );
document.write( "This works because R(x) is a quadratic function that graphs out a parabola. This parabola opens downward producing a highest point at the vertex (h,k). This is where the revenue is maxed out.\r<BR>\n" );
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document.write( "A different way to find the x coordinate of the vertex is to locate the roots of R(x). Let's say the roots are p and q. The average of the roots is exactly the value of h. This is due to symmetry. So we can say h = (p+q)/2.\r<BR>\n" );
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document.write( "Whichever method you use to find the x coordinate of the vertex, plug that x value into the revenue function to get<BR>\n" );
document.write( "R(x) = (40000-800x)*(200+10x)<BR>\n" );
document.write( "R(15) = (40000-800*15)*(200+10*15)<BR>\n" );
document.write( "R(15) = (40000-12000)*(200+150)<BR>\n" );
document.write( "R(15) = (28000)*(350)<BR>\n" );
document.write( "R(15) = 9,800,000<BR>\n" );
document.write( "The weekly revenue is maxed out at 9.8 million dollars when they increase the ticket price a total of 15 times. Each increase is by $10 increments.\r<BR>\n" );
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document.write( "Now the question is what should the ticket price be. <BR>\n" );
document.write( "Well that's the result of 200+10x when x = 15, so,<BR>\n" );
document.write( "ticket price = 200+10x<BR>\n" );
document.write( "ticket price = 200+10*15<BR>\n" );
document.write( "ticket price = 200+150<BR>\n" );
document.write( "ticket price = 350 dollars\r<BR>\n" );
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document.write( "You were fairly close when you got 300 dollars. However, I'm not quite sure how you got those steps.\r<BR>\n" );
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document.write( "<font color=red>Answer: 350 dollars</font></font> </SPAN>\n" );
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