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.
What should she charge to maximize her revenue?
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<pre>
After increasing the price n times, the price per ticket is 100+3n dollars,
but the number of sold tickets is 2000-20n.


Thus, after changing the price n times, the revenue is

      R(n) = (2000-20n)*(100+3n) = 200000 - 2000n + 6000n - 60n^2 = -60n^2 + 4000n + 200000.    (1)


We want to find "n" in a way to maximize quadratic function (1).


   +-------------------------------------------------------------------+
   |       The maximum is at  n = " {{{-b/(2a)}}} ",  where "a"               |
   |    is the coefficient at n^2 and "b" is the coefficient at n.     |
   +-------------------------------------------------------------------+


In our problem,  a= -60,  b= 4000,  so the desired n is 

    n = {{{ -4000/(2*(-60))}}} = {{{4000/120}}} = {{{400/12}}} = {{{100/3}}} = 33{{{1/3}}}.



Next, the number of n in this problem/solution not necessary should be integer,

but the numbers 100+3n and 2000-3n must be integer numbers, due to their meaning.

         (100+3n must be integer number of cents, at least.)



We have 100+3n = 200  is still integer,  but 200-20n  is not integer at n= 33{{{1/3}}}.


THEREFORE, we should test the closest integers n= 33 and n= 34 as possible candidates.


We have  the revenue R(33) = -60*33^2 + 4000*33 + 200000 = 266660 dollars,

         the revenue R(34) = -60*34^2 + 4000*34 + 200000 = 266640 dollars.


The revenue R(33) is higher, so we choose n= 33 as the number of increases.


It gives the optimum price  100 + 33*3 = 199 dollars.


<U>ANSWER</U>.  Optimum price is 199 dollars per ticket.
</pre>

Solved.


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To learn about finding the minimum/maximum of a quadratic function, &nbsp;see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>

in this site.



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic &nbsp;"<U>Finding minimum/maximum of quadratic functions</U>". 



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.



I advise you to look into many other lessons in this topic - you will find there 

a lot of similar problems solved - - - to make your horizon wider.