A baseball team plays in a stadium that holds 70000 spectators. With the ticket price at $11 the average attendance has been 30000. When the price dropped to $9, the average attendance rose to 35000.
a) Find the demand function p(x), where x is the number of the spectators. (Assume p(x) is linear.)
To find the demand function, we need to determine the relationship between the ticket price and the average attendance.
Let's first determine the slope of the demand function. The slope represents the change in average attendance divided by the change in ticket price. We have two points to work with:
Point 1: ($11, 30000)
Point 2: ($9, 35000)
Change in ticket price (Δp) = $9 - $11 = -$2
Change in average attendance (Δx) = 35000 - 30000 = 5000
Slope (m) = Δx / Δp = 5000 / (-$2) = -2500
Now that we have the slope, we can use it to find the demand function. The general equation for a linear demand function is p(x) = mx + b, where m is the slope and b is the y-intercept.
Using one of the points, ($11, 30000), we can substitute its values into the equation to solve for the y-intercept, b.
$11 = -2500(30000) + b
$11 = -75000000 + b
b = $11 + $75000000
b = $75000011
Therefore, the demand function p(x) is:
p(x) = -2500x + $75000011