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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?

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.

### 1 Answer by Expert Tutors

Andrew M. | Mathematics - Algebra a Specialty / F.I.T. Grad - B.S. w/HonorsMathematics - Algebra a Specialty / F.I....
0
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