document.write( "Question 1196777: Alex is selling tickets. If she charges $100 she can sell 2000 tickets. For every $3 increase in price she sells 20 fewer tickets.
\n" ); document.write( "What should she charge to maximize her revenue?
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #829762 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Alex is selling tickets. If she charges $100 she can sell 2000 tickets.
\n" ); document.write( "For every $3 increase in price she sells 20 fewer tickets.
\n" ); document.write( "What should she charge to maximize her revenue?
\n" ); document.write( "~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "After increasing the price n times, the price per ticket is 100+3n dollars,\r\n" );
document.write( "but the number of sold tickets is 2000-20n.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus, after changing the price n times, the revenue is\r\n" );
document.write( "\r\n" );
document.write( "      R(n) = (2000-20n)*(100+3n) = 200000 - 2000n + 6000n - 60n^2 = -60n^2 + 4000n + 200000.    (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We want to find \"n\" in a way to maximize quadratic function (1).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "   +-------------------------------------------------------------------+\r\n" );
document.write( "   |       The maximum is at  n = \"  \",  where \"a\"               |\r\n" );
document.write( "   |    is the coefficient at n^2 and \"b\" is the coefficient at n.     |\r\n" );
document.write( "   +-------------------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In our problem,  a= -60,  b= 4000,  so the desired n is \r\n" );
document.write( "\r\n" );
document.write( "    n =  =  =  =  = 33.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, the number of n in this problem/solution not necessary should be integer,\r\n" );
document.write( "\r\n" );
document.write( "but the numbers 100+3n and 2000-3n must be integer numbers, due to their meaning.\r\n" );
document.write( "\r\n" );
document.write( "         (100+3n must be integer number of cents, at least.)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We have 100+3n = 200  is still integer,  but 200-20n  is not integer at n= 33.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "THEREFORE, we should test the closest integers n= 33 and n= 34 as possible candidates.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We have  the revenue R(33) = -60*33^2 + 4000*33 + 200000 = 266660 dollars,\r\n" );
document.write( "\r\n" );
document.write( "         the revenue R(34) = -60*34^2 + 4000*34 + 200000 = 266640 dollars.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The revenue R(33) is higher, so we choose n= 33 as the number of increases.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It gives the optimum price  100 + 33*3 = 199 dollars.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  Optimum price is 199 dollars per ticket.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To learn about finding the minimum/maximum of a quadratic function, see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this textbook under the topic \"Finding minimum/maximum of quadratic functions\". \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I advise you to look into many other lessons in this topic - you will find there \r
\n" ); document.write( "\n" ); document.write( "a lot of similar problems solved - - - to make your horizon wider.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );