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Quadratic functions play a significant role in understanding revenue optimization, especially in this motel rental problem. A quadratic function typically takes the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The revenue function derived in this exercise, \( R(n) = 87n - 2n^2 \), is a quadratic function. Why is this relevant? Quadratic functions create a parabola when graphed, and when the leading coefficient \( a \) is negative (as in \( -2 \)), the parabola opens downward.
This means the revenue function has a maximum point, or vertex, which helps identify the optimal number of rooms to rent for maximizing revenue.
By locating the vertex of this parabola, we find the peak revenue potential. The vertex can be found using the formula \( n = -\frac{b}{2a} \), where \( b \) and \(a \) are taken from the quadratic function. In this exercise, by substituting the values \( b = 87 \) and \( a = -2 \), we calculated the optimal \( n \), resulting in the most profitable number of room rentals. Understanding these properties of quadratic functions is crucial when maximizing revenue or optimizing other outcomes in similar situations.

Algebraic modeling involves the use of algebraic equations to represent real-world scenarios, allowing for analysis and problem-solving. In this case, algebraic modeling is employed to represent the pricing and revenue structure of the motel’s room rentals.
By using the information provided, we derived equations that not only depict the cost per room but also the total revenue from renting multiple rooms. The initial step was understanding the pricing decrease associated with each additional room, modeled by the equation for the rate per room: \( 87 - 2n \).
This equation models the relationship between the number of rooms rented, \( n \), and the price per room, leading to the main equation for calculating total revenue, \( R(n) = 87n - 2n^2 \). Such equations simplify complex scenarios into manageable calculations, making it easier to predict and analyze outcomes. With algebraic modeling, businesses can make informed decisions about pricing strategies, resource allocation, and optimal service offerings.

Pricing strategies are essential for maximizing revenue, particularly in competitive markets like hospitality. The pricing strategy employed by the motel in this scenario uses a decremental pricing model, where the rate per room decreases with each additional room rented by a guest.
This approach encourages customers to rent more rooms by offering a bulk discount, a common strategy to increase overall occupancy and revenue. The understanding and application of a decremental pricing model can profoundly affect business profitability. The derived formula \( 87 - 2n \) for pricing structure highlights how prices are adjusted based on the volume of services consumed.
This strategy needs to account for optimal pricing that maximizes revenue without severely diminishing the value of each room. By strategically setting rates, businesses can increase their allure to customers who are price-sensitive and likely to opt for value deals. This exercise allows the motel to balance the dual objectives of maximizing income and ensuring customer satisfaction through nuanced pricing strategies.