Find the demand function p ( x ) , where x is the number of the spectators. (Assume p ( x ) is linear.

Question

A baseball team plays in he stadium that holds 60000 spectators. With the ticket price at $11 the average attendence has been 23000. When the price dropped to $9, the averege attendence rose to 30000.

Find the demand function , where  is the number of the spectators. (Assume is linear.)

Answer 1

Standard form of a linear demand equation is Q = a - bP.

Q = quantity, a = factors beside price, b = slope, P = price

If we have two prices/crowds, we will get simultaneous equations to solve.

Substitute (23000, 11) to get (1) 23000 = a - 11b

Substitute (30000, 9) to get (2) 30000 = a - 9b

subtract (2) from (1) to eliminate a gives -7000 = -2b

=> b = 3500

substitute b value to get 23000 = a - 11(3500)

=> a = 23000 + 11(3500) = 61500

Hence the demand function is Q = 61500 - 3500P

But we want Price given Q so rearrange to get P = ( 61500 - Q ) / 3500

Hence our required price function is P(x) = ( 61500 - x) / 3500

To double check, we can try it with crowds of 23,000 and 30,000 and a full house of 60,000 to get $11, $9, and 43 cents!