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Question

A baseball team plays in a stadium that holds 56,000 spectators. With the ticket price at $10, the average attendance at recent games has been 30,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000.

(a) Find a function that models the revenue in terms of ticket price. (Let x represent the price of a ticket and R represent the revenue.) 

R(x) =

(b) Find the price that maximizes revenue from ticket sales. 
$  

(c) What ticket price is so high that no revenue is generated? 
$

Answer 1

We can relate the attendance to the ticket price as a=mx+c where m and c are to be found.

When x=10 a=30000 and when x=9 a=33000.

Plugging these values in we get 30000=10x+c and 33000=9x+c, so if we subtract these two equations from one another we eliminate c: x=-3000. Now we can calculate c: 30000=-30000+c, so c=60000 and a=60000-3000x. We have related attendance to the ticket price.

R(x)=ax.

Now we can relate the revenue to the ticket price: (a) R(x)=60000x-3000x².

(b) R(x)=-3000(x²-20x+100-100)=-3000(x-10)²+300000. When x=$10 R(x) is maximum.

(c) R(x)=0 when 3000x=60000, so x=$20 (and attendance is zero).