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Express Lane Plus
Posts: 485
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4 weeks ago
A baseball team plays in a stadium that holds 64000 spectators. With the ticket price at $9 the average attendance has been 29000. When the price dropped to $7, the average attendance rose to 32000. Assume that attendance is linearly related to ticket price.
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Educator
4 weeks ago
Let x = the ticket cost reduction in $.  The ticket price function will be:
 
Price = $9 - x
 
The demand function (number of spectators) is:
 
p(x) = 27,000 + 5000x
 
The revenue is:
 
R(x) = (Demand)(Price)
R(x) = (29,000 + 5000x)(9 - x) = 261,000 + 45,000x + 29,000x - 5,000x^2
R(x) = \(261,000+74,000x-5,000x^2\)
 
To find the optimal revenue, take the derivative of R(x) wrt x, set it to zero, and solve for x.

\(R'(x)=74,000-10,000x\)

\(-74,000=-10,000x\)

\(x=\frac{-74}{-10}=7.4\)

Price = $9 - 7.4.  Answer is $7.60.
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