calculus homework

Question

A baseball team plays in a stadium that holds 54,000 spectators. With ticket prices at $10, the average attendance had been 16,000. When ticket prices were lowered to $8, the average attendance rose to 24,000.



(a) Find the demand function (price as a function of attendance), assuming it to be linear.

p(x) =  

 

 

(b) How should ticket prices be set to maximize revenue?

$ 

Answer 1

(16,10), (24,8) are two points on a line, measured in units of thousands and dollars

slope = -1/4 = (10-8)/(16-24)

p(x) = -x/4 +14

10 =-16/4 +14

0 = -56/4 +14

P intercept = 14 or (0, 14)

x intercept = 56 or (56,0)

max revenue is the midpoint between the intercepts (56/2, 14/2) = (28,7) with $7 = Price and 28,000 seats

max revenue = 7(28000) = $196,000

that's a geometric solution

or there's calculus:

R(x) = px = -x^2/4 +14x

R'(x) = -x/2 +14 = 0

x = 28

P = -x/4 + 14 = -28/4+14= 7