Question

Community Theater Owner A community theater currently sells 200 season tickets at $50 each.
In order to increase its season-ticket revenue, the theater surveys its season ticket holders to see if they would be willing to pay more.
The survey finds that for every $5 increase in the price of a season ticket, the theater would lose 10 season-ticket holders.
What action, if any, should the theater owner take to increase revenue?
a. Let n be the number of $5 price increases in the cost of a season ticket. Write an expression for the cost of a season ticket after n price increases, and an expression for the number of season-ticket holders after n price increases.
b. Use the expressions from part a to create a revenue function, R (n) , from the survey information.
c. Graph the revenue function. Be sure to label the axes with the quantities they represent and indicate the axis scales by showing numbers for some grid lines.
e. Write a brief paragraph describing what actions the theater owner should take to maximize revenue. Include what happens to the number of season-ticket holders as well as the season-ticket prices.

Answer 1

Given :  A community theater currently sells 200 season tickets at $50 each.  The survey finds that for every $5 increase in the price of a season ticket, the theater would lose 10 season-ticket holders.

To find :  Revenue Function , maximum revenue , revenue graph

Solution:

200 season tickets at $50 each.

Revenue = 200 * 50  = $ 10000

cost of a season ticket  = 50 + 5n

number of season-ticket holders = 200 - 10n

Revenue (= (200 - 10n) (50 + 5n)

= 10000 + 1000n - 500n  - 50n²

= 10000 + 500n  - 50n²

= 50 ( 200 + 10n - n²)

R(n) = 50 ( 200 + 10n - n²)

dR/dn  =  50(10 - 2n)

=> 2n = 10

=> n = 5

d²R/dn² = 50(-2) = - 100 < 0

Hence  maximum revenue at n = 5

maximum Revenue = (200 - 10*5) (50 + 5*5)  

= 150 * 75

= 11250 $

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