Problem 50 - Algebra for College Students

Question

Motel managers advertise that they will provide dinner, dancing, and drinks for \(\$ 50\) per couple for a New Year's Eve party. They must have a guarantee of 30 couples. Furthermore, they will agree that for each couple in excess of 30 , they will reduce the price per couple by \(\$ 0.50\) for all attending. How many couples will it take to maximize the motel's revenue?

Step-by-Step Solution

Answer

The motel maximizes revenue with 65 couples attending.

Step 1: Define the Revenue Function

The motel earns revenue from each couple based on the pricing structure. Let \( x \) represent the number of additional couples beyond the 30-couple guarantee. The price per couple decreases by \( \$0.50 \) for each additional couple, so the price per couple is \( 50 - 0.5x \). The total number of couples is \( 30 + x \). The revenue \( R \) is the product of the number of couples and the price per couple: \( R(x) = (30 + x)(50 - 0.5x) \).

Step 2: Expand the Revenue Function

To maximize the revenue, we need to expand the revenue function \( R(x) = (30 + x)(50 - 0.5x) \). Expanding, we get:\[ R(x) = 30 \cdot (50 - 0.5x) + x \cdot (50 - 0.5x) = 1500 - 15x + 50x - 0.5x^2 \]. Simplifying, we have: \[ R(x) = 1500 + 35x - 0.5x^2 \].

Step 3: Convert to a Quadratic Function

The expanded equation \( R(x) = 1500 + 35x - 0.5x^2 \) is a quadratic function in the standard form \( R(x) = ax^2 + bx + c \). Here, \( a = -0.5 \), \( b = 35 \), and \( c = 1500 \). Since \( a < 0 \), the parabola opens downward, indicating that there is a maximum point.

Step 4: Find the Vertex of the Parabola

The vertex of a parabola in the form \( ax^2 + bx + c \) represents the maximum point for \( a < 0 \). The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Substituting into the formula, we have:\[ x = -\frac{35}{2(-0.5)} = \frac{35}{1} = 35 \].

Step 5: Calculate Total Number of Couples

Since \( x \) represents the number of additional couples beyond the 30 guaranteed, the total number of couples is \( 30 + x = 30 + 35 = 65 \). Thus, 65 couples will maximize revenue.

Key Concepts

Quadratic Functions

Quadratic functions are an essential concept in algebra, especially when modeling situations involving revenue maximization. Typically, a quadratic function is expressed in the form \( ax^2 + bx + c \). These functions describe parabolic graphs that can either open upwards or downwards. In this article, we'll understand how to apply quadratic functions within the context of a practical situation: maximizing revenue at a motel event.
In our scenario, the revenue function \( R(x) = 1500 + 35x - 0.5x^2 \) is a quadratic function with a negative leading coefficient \( a = -0.5 \). This means the parabola opens downward, indicating there's a maximum revenue point. Understanding this structure helps in analyzing how changes in price and attendance influence total revenue.

Vertex of a Parabola

The vertex of a parabola is a crucial concept when working with quadratic functions, especially those aimed at optimization problems like maximizing revenue. The vertex serves as the peak (or lowest point) of the parabola. For a quadratic function \( ax^2 + bx + c \), the vertex gives us the maximum or minimum value of the function.
In our example, the revenue function's vertex is found using the formula \( x = -\frac{b}{2a} \). Substituting in our values \( b = 35 \) and \( a = -0.5 \), we find \( x = 35 \). This calculation helps us determine the number of additional couples needed beyond the guaranteed couples to achieve maximum revenue. This outcome is significant for decision-making in practical scenarios.

Revenue Function

A revenue function is a vital mathematical tool used to model income based on certain variables. It's particularly useful in economic and business contexts to determine the amount of money generated under specific conditions.
In the case of the motel event, the revenue function \( R(x) = (30 + x)(50 - 0.5x) \) models how changes in the number of couples and pricing affect total revenue. Breaking down this model, \( 30 + x \) represents the total number of couples, while \( 50 - 0.5x \) accounts for the price per couple. By expanding and simplifying this expression, we gain a clearer picture of how different factors interact to influence overall income.

Mathematical Modeling

Mathematical modeling is a powerful method used to represent real-world situations through mathematical formulas and expressions. This approach allows us to interpret, analyze, and predict outcomes based on given data and variables.
In our motel example, mathematical modeling helps in understanding how adjusting prices with changing attendance impacts revenue. By developing a revenue function, managers can predict optimal pricing and attendance numbers. This modeling provides clear guidance on how to structure events to ensure maximum financial gain. It's a crucial skill that merges algebraic principles with real-world decision-making processes, allowing stakeholders to make well-informed choices.