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A small theater has a seating capacity of 2000. When the ticket price is $20, attendance is 1500. For each $1 decrease in price, attendance increases by 100.

(a) Write the revenue R of the theater as a function of ticket price x.
R(x) =


(b) What ticket price will yield a maximum revenue?
$


What is the maximum revenue?

Comments

Revenue = Price*attendance
r(x) = xa
 
Price       Attendance       Revenue
_______________________________
20            1500               $30,000
19            1600               $30,400
18            1700               $30,600
17            1800               $30,600
16            1900               $30,400
15            2000               $30,000
 
Note that no information is given as to whether or not the
ticket sales must be cut in $1 increments.  Example, if we
sell the tickets for $17.50 do we get 1750 attendees?
This would be at the vertex of the parabolic curve indicating
a maximum revenue.
17.5(1750) = $30,625 revenue.
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1 Answer

Letting x represent the change in ticket price in dollars:
 
R(x) = (20-x)(1500 + 100x); 0≤ x ≤ 5   x cannot go higher than 5
                                                           because this corresponds to selling
                                                           out the 2,000 seat theater.
 
R(x) = 30000 + 2000x - 1500x - 100x2
R(x) = -100x2 + 500x + 30000
 
Find the vertex of the parabolic function:
Vertex:  (-b/2a, r(-b/2a)); a = -100, b=500
 
-b/2a = -500/-200 = 2.5
 
r(2.5) = -100(2.5)2 + 500(2.5) + 30000
          = -625 + 1250 + 30000
          = $30,625
 
The maximum revenue is at the point where ticket prices are
reduced by $2.5 to $17.50.   The revenue at that point is $30,625