Problem 9 - Precalculus Essentials

Question

A football team plays in a large stadium. With a ticket price of \(\$ 20,\) the average attendance at recent games has been 30,000 A market survey indicates that for each \(\$ 1\) increase in the ticket price, attendance decreases by 500 . a. Express the number of spectators at a football game, \(N\), as a function of the ticket price, \(\bar{x}\). b. Express the revenue from a football game, \(R\), as a function of the ticket price, \(x\)

Step-by-Step Solution

Answer

The number of spectators at a football game, \(N\), as a function of the ticket price, \( \bar{x} \) is: \( N = 40000 - 500\bar{x} \). And the revenue from a football game, \(R\), as a function of the ticket price, \( \bar{x} \) is: \( R = 40000\bar{x} - 500\bar{x}^2 \)

Step 1: Define the spectators function

Using the given, start at the initial point where the ticket price \( \bar{x} \) is $20, and the number of spectators \( N \) is 30,000. Each $1 increase in \( \bar{x} \) results in a decrease of 500 in \( N \). So, if \( \bar{x} \) increases by \( d \), \( N \) decreases by \( 500d \). Thus, \( N \) can be expressed as a function of \( \bar{x} \) as follows: \( N = 30000 - 500(\bar{x} - 20) \)

Step 2: Simplify the spectators function

To make the spectators function simpler, distribute the 500 to get: \( N = 30000 - 500\bar{x} + 10000 \), which simplifies to: \( N = 40000 - 500\bar{x} \)

Step 3: Define the revenue function

The revenue \( R \) is the product of the number of spectators \( N \) and the ticket price \( \bar{x} \). So, \( R \) can be expressed as a function of \( \bar{x} \) using the \( N \) function derived earlier: \( R = \bar{x}(40000 - 500\bar{x}) \)

Step 4: Simplify the revenue function

To make the revenue function simpler and easier to understand, expand the equation: \( R = 40000\bar{x} - 500\bar{x}^2 \), which is a quadratic function in terms of \( \bar{x} \)

Key Concepts

Price Elasticity of Demand

Price elasticity of demand is a measurement that quantifies how the quantity demanded of a good responds to a change in its price. Precisely, it's the percentage change in quantity demanded in response to a one percent change in price. Understandably, different products have different elasticity. For instance, essential items often have low elasticity because people will buy them regardless of price changes, while non-essential luxuries may have high elasticity.In our football team scenario, for each $1 increase in ticket price, 500 fewer people attend the game. This helps to understand the sensitivity of the fans' attendance to ticket price changes. When the price goes up, the number attending goes down, suggesting a negative elasticity, which is typical for most goods and services. The elasticity can help business or team owners predict how changes in pricing might affect their sales volume and revenue.

Revenue Optimization

Revenue optimization is about finding that sweet spot in pricing that maximizes income. It's a crucial consideration for any organization. As we saw in our football team's example, increasing the price of tickets impacts attendance. The goal is to determine the ideal ticket price that brings in the most revenue without dissuading too many fans.In mathematical terms, we seek to maximize the revenue function, which is often a quadratic equation due to the linear relationship between price change and demand. Revenue optimization requires analyzing how changes in price will affect total revenue, which involves pricing strategies, understanding consumer behavior, and using mathematical models to predict outcomes.

Quadratic Functions

Quadratic functions are polynomial functions of the second degree, typically taking the form of \( ax^2 + bx + c \), where \((a, b, c)\) are constants, and \((a \eq 0)\). These functions create a parabola when graphed on a coordinate plane, with a distinct vertex that represents either the maximum or minimum value.In the context of our football ticket problem, the revenue function simplifies to a quadratic function, \( R = 40000\bar{x} - 500\bar{x}^2 \), reflecting the nonlinear impact of ticket price changes on revenue. By studying the properties of quadratic functions, one can determine the best pricing strategy to maximize revenue. The concept of the quadratic function is pivotal in many areas of precalculus, and understanding how to graph and analyze these functions is vital for solving real-world problems.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world situations. This enables us to analyze and make predictions about complex systems. In the case of a football stadium's ticket sales, we use mathematical modeling to express the relationships between ticket price, attendance, and revenue.By translating the given information into equations, we can create a model that predicts how many spectators will attend a game at various ticket prices and, subsequently, how much revenue will be generated. In this scenario, our model takes the form of a quadratic function, which is typical for modeling situations with maximum or minimum values, like revenue optimization in this case. The strength of mathematical modeling lies in its ability to simplify and solve for the unknowns in real-world problems, just as we've done with the football team's ticket pricing strategy.