Calculus Optimization Question

Question

A baseball team plays in a stadium that holds 50000 spectators. With the ticket price at $8 the average attendance has been 19000. When the price dropped to $6, the average attendance rose to 25000.

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

Answer 1

Consider it as two points on a graph

with (x,p) = (19,8) and (25,6). The slope of that line is -1/3

use the point slope formula to find the equation of the line, or the demand function

p-8 = (-1/3)(x-19)

p(x) = -x/3 +19/3 + 8 = -x/3 + 43/3

p(x) = -x/3 + 43/3

if p = 0, then x/3 = 43/3 and x= attendance of 43,000, the maximum unless you made price negative and paid people to come, until you reached the maximum capacity of 50,000

p(19) = -19/3 +19/3 + 8 = $8 when 19 thousand attend

p(25) = -25/3 +19/3 + 8 = -6/3 + 8 = $6 when 25 thousand attend

charge p=$14.33 and no one will buy a ticket