A baseball team plays in a stadium that holds 66000 spectators. With the ticket price at $12 the average attendence has been 29000. When the price dropped to $9, the average attendence rose to 33000. Assume that attendence is linearly related to ticket price.What ticket price would maximize revenue? Please help ASAP!!! :(
To determine the ticket price that would maximize revenue, we need to find the price at which the product of the ticket price and the attendance is highest. Let's calculate the revenue at each ticket price to determine the maximum.
Revenue is calculated as the product of the ticket price and the attendance.
1. Calculate the revenue when the ticket price is $12:
Revenue at $12 = $12 × 29,000 = $348,000
2. Calculate the revenue when the ticket price is $9:
Revenue at $9 = $9 × 33,000 = $297,000
To maximize revenue, we need to compare the revenue at different ticket prices. In this case, $12 generated higher revenue than $9, so we know the optimal ticket price is greater than $12.
To continue narrowing down the optimal ticket price, we can analyze the trend between revenue and ticket price.
The attendance is linearly related to the ticket price. As the ticket price drops from $12 to $9, attendance increases from 29,000 to 33,000. This means there is an increase of 4,000 attendees for every $3 decrease in ticket price.
To find the optimal ticket price, we need to determine the ticket price that generates the maximum revenue. We can assume that this trend continues and calculate the revenue at a lower ticket price.
3. Calculate the revenue at a lower ticket price:
Revenue at $6 = $6 × (33,000 + 4,000) = $222,000
From the calculations, we can see that reducing the ticket price to $6 decreases the revenue. Therefore, the optimal ticket price for maximizing revenue is between $9 and $12.
To further narrow it down, we can calculate the revenue at a ticket price between $9 and $12. Let's check the revenue at $10.50:
Revenue at $10.50 = $10.50 × (29,000 + 2,000) = $315,000
Comparing the revenues, we see that $10.50 generates more revenue than both $9 and $12.
Therefore, the ticket price that maximizes revenue is $10.50.
To determine the ticket price that would maximize revenue, we need to understand the relationship between ticket price and attendance, and how it impacts the revenue generated.
From the given information, we know that the attendance is linearly related to the ticket price. Let's denote the ticket price as "x" (in dollars) and the attendance as "y" (number of spectators).
We have two data points:
Point 1: Ticket price = $12, Attendance = 29000
Point 2: Ticket price = $9, Attendance = 33000
Using these two points, we can calculate the slope of the attendance function:
Slope (m) = (Change in attendance) / (Change in ticket price)
= (y2 - y1) / (x2 - x1)
= (33000 - 29000) / ($9 - $12)
= 4000 / -$3
= -4000/3
Now that we know the slope of the attendance function, we can express it in terms of "x" and "y":
Attendance = -4000/3 * (x - x1) + y1
To maximize revenue, we need to find the ticket price that yields the maximum attendance and then calculate the revenue.
Since the stadium capacity is 66000, the maximum attendance cannot exceed this value. Let's determine the maximum attendance:
66000 = -4000/3 * (x - x1) + y1
Solving for x, we get:
x = (66000 - y1) * (-3/4000) + x1
Substituting the values of y1 = 29000 and x1 = $12 into the equation:
x = (66000 - 29000) * (-3/4000) + $12
= 37000 * (-3/4000) + $12
= -277.5 + $12
= $-265.5
It doesn’t make sense to have a negative ticket price, so we need to consider the other data point to find the ticket price that maximizes revenue.
Now let's calculate the revenue at the given data points:
Revenue = Ticket price * Attendance
Revenue at Point 1: $12 * 29000 = $348,000
Revenue at Point 2: $9 * 33000 = $297,000
Comparing the two revenue values, we can see that the revenue is maximized at Point 1, given a ticket price of $12.
Therefore, the ticket price that would maximize revenue is $12.
the attendance rose 4000 on a price drop of $3
So, if there are x price drops, the revenue R(x) will be
R(x) = (29000+4000x)(12-3x)
Now just find the vertex of that parabola.
Make sure that it does not occur when attendance is greater than 66,000