A baseball team plays in a stadium that holds 74000 spectators. With the ticket price at $11, the average attendance has been 30000. When the price dropped to $8, the average attendance rose to 37000. Assume that attendance is linearly related to the ticket price.
What ticket price would maximize revenue?
The attendance is a straight line through the two points (11,30) and (8,37)
a = -7/3 (t-8)+37
Revenue = price * attendance
r(t) = t(-7/3 (t-8)+37) = 1/3 (167t-7t^2)
The vertex is at t = -b/2a = 167/14 = $11.93
To find the ticket price that would maximize revenue, we need to determine the price at which the total revenue is the highest. Total revenue is calculated by multiplying the ticket price by the number of attendees.
Let's start by calculating the revenue at the initial ticket price of $11.
Revenue at $11 = Price * Average Attendance
Revenue at $11 = $11 * 30,000
Now, let's calculate the revenue at the reduced ticket price of $8.
Revenue at $8 = Price * Average Attendance
Revenue at $8 = $8 * 37,000
To determine the linear relationship between attendance and ticket price, we can use the formula for a line, y = mx + b, where y represents the average attendance, x represents the ticket price, m represents the slope, and b represents the y-intercept.
Using the given data points (30000, 11) and (37000, 8), we can calculate the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (8 - 11) / (37000 - 30000)
Once we have the slope, we can derive the equation for the linear relationship between attendance and ticket price.
Equation: y = mx + b
Average Attendance = Slope * Ticket Price + y-intercept
We already have the slope (m) from the earlier calculation. Now, we need to determine the y-intercept (b) by substituting one of the data points into the equation:
30000 = m * 11 + b
From this equation, we can find the value of b.
Now, we can solve for the revenue at any ticket price using the equation y = mx + b.
To maximize revenue, we need to find the ticket price that corresponds to the maximum revenue value. We can do this by finding the vertex of the revenue function, which represents the maximum point.
The vertex formula for a parabolic equation is:
x = -b / (2a)
Since the attendance and ticket price have a linear relationship, the revenue function forms a linear relationship as well. Hence, there is no parabola, and we can simply maximize the revenue by finding the ticket price that generates the highest revenue.
By comparing the revenues at different ticket prices, we can determine the ticket price that maximizes revenue.
To find the ticket price that would maximize revenue, we need to determine the point at which the revenue is highest.
The revenue can be calculated by multiplying the ticket price by the attendance. So, revenue = ticket price * attendance.
First, let's calculate the revenue at the $11 ticket price. The average attendance is 30,000 spectators, so the revenue at $11 is:
Revenue at $11 = $11 * 30,000 = $330,000
Next, let's calculate the revenue at the $8 ticket price. The average attendance is 37,000 spectators, so the revenue at $8 is:
Revenue at $8 = $8 * 37,000 = $296,000
To find the ticket price that maximizes the revenue, we need to find the point where the revenue is highest. The revenue is increasing as the ticket price decreases. Therefore, we need to find the price that gives the highest revenue.
We can plot a graph with the ticket price on the x-axis and the revenue on the y-axis. The revenue will increase as the ticket price increases, so we can draw a line connecting the two points we have: ($11, $330,000) and ($8, $296,000).
Now, we need to find the midpoint of the line connecting these two points. This midpoint will represent the ticket price that maximizes the revenue.
Using the midpoint formula, we can calculate the x-coordinate (ticket price) of the midpoint:
Midpoint = ( ($11 + $8) / 2 , (30,000 + 37,000) / 2 )
= ($19/2, 67,000/2)
= ($9.5, 33,500)
Therefore, the ticket price that would maximize revenue is $9.5.