Problem 77 - Algebra and Trigonometry

Question

Stadium Revenue A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price at $10, the average attendance at recent games has been 27,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000. (a) Find a function that models the revenue in terms of ticket price. (b) Find the price that maximizes revenue from ticket sales. (c) What ticket price is so high that no revenue is generated?

Step-by-Step Solution

Answer

a) \( R(x) = 57,000x - 3,000x^2 \); b) Max revenue at $9.5; c) $19 means zero revenue.

Step 1: Define Variables and Initial Conditions

Let the ticket price be \( x \) dollars. Currently, with \( x = 10 \), the attendance is 27,000. When the price decreases by $1, attendance increases by 3,000 spectators.

Step 2: Establish Attendance Function

The attendance \( A(x) \) in terms of ticket price \( x \) is given by: \( A(x) = 27,000 + 3,000(10 - x) = 57,000 - 3,000x \). This is because the price decrease causes an increase in attendance.

Step 3: Determine Revenue Function

Revenue \( R(x) \) is calculated by multiplying ticket price \( x \) by attendance \( A(x) \): \[ R(x) = x(57,000 - 3,000x) = 57,000x - 3,000x^2 \]. This is a quadratic function.

Step 4: Find the Price That Maximizes Revenue

The revenue function \( R(x) = 57,000x - 3,000x^2 \) is a downward-opening parabola. Its maximum value can be found at the vertex. The vertex \( x \) is given by \( x = -\frac{b}{2a} \), where \( a = -3,000 \) and \( b = 57,000 \).\[ x = -\frac{57,000}{2(-3,000)} = 9.5 \]. Thus, the ticket price that maximizes revenue is $9.5.

Step 5: Identify Price Leading to Zero Revenue

For zero revenue, the attendance should reduce to zero, i.e., \( A(x) = 0 \). Hence, solve \( 57,000 - 3,000x = 0 \).\[ 3,000x = 57,000 \Rightarrow x = 19 \]. Therefore, the ticket price at which the revenue becomes zero is $19.

Key Concepts

Quadratic Functions

Quadratic functions play a crucial role in many mathematical modeling scenarios, especially those involving optimization like revenue optimization. A quadratic function is typically expressed as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In this context, the quadratic function models revenue based on ticket price.
For our stadium exercise, the revenue function is \( R(x) = 57,000x - 3,000x^2 \). This function is downward-opening because the coefficient of \( x^2 \) is negative, indicating that the graph is a parabola that opens downwards.

Key characteristics of quadratic functions include:

• The vertex, which provides the maximum or minimum value of the function.
• The roots or x-intercepts where the function equals zero.

In our exercise, finding the vertex tells us the optimal ticket price for maximizing revenue, while solving for roots shows us when revenue drops to zero.

Attendance Modeling

Attendance modeling is an essential concept for understanding how changes in variables, like ticket prices, influence attendance at events. This exercise showcases a real-life scenario where ticket price impacts spectator numbers in a stadium.
The function \( A(x) = 57,000 - 3,000x \) models attendance based on ticket price \( x \). Here, 57,000 is the potential maximum attendance when tickets are free, and 3,000 represents the rate at which attendance decreases as ticket prices rise by $1.

Advantages of such a model include:

• It captures the relationship between ticket price and attendance.
• It makes predictions about future attendance based on price changes.

Understanding attendance models helps optimize pricing strategies for maximizing total revenue.

Ticket Pricing Strategy

A ticket pricing strategy is pivotal for balancing revenue with attendance. By setting the right price, teams can maximize their earnings while ensuring stadium capacity isn't exceeded. This problem demonstrates how a slight adjustment in ticket prices can significantly impact overall revenue.
The main goal is to find that sweet spot where the ticket price maximizes revenue without discouraging too much attendance. From the quadratic revenue function \( R(x) = 57,000x - 3,000x^2 \), the price that achieves this is identified using the formula for the vertex of a parabola, highlighting $9.5 as the desired price point.

Important considerations in ticket pricing strategy include:

• Laws of supply and demand.
• Market survey results that quantify audience responsiveness.

By carefully modeling these factors, organizations can create effective strategies that optimize both attendance and profits.