Problem 58 - University Calculus: Early Transcendentals

Question

The 800 -room Mega Motel chain is filled to capacity when the room charge is \(\$ 50\) per night. For each \(\$ 10\) increase in room charge, 40 fewer rooms are filled each night. What charge per room will result in the maximum revenue per night?

Step-by-Step Solution

Answer

The charge for maximum revenue is $125.

Step 1: Define Variables

Let the room charge be \( x \) dollars per night. The current charge is \( 50 \) dollars, and each \( 10 \) dollars increase results in 40 fewer rooms filled.

Step 2: Determine the Price and Number of Rooms Relationship

If \( y \) is the number of \( 10 \) dollar increases, the new room charge is \( 50 + 10y \). The number of rooms filled is \( 800 - 40y \).

Step 3: Establish Revenue Equation

Revenue \( R(y) \) is the product of the room charge and the number of rooms filled, so \( R(y) = (50 + 10y)(800 - 40y) \).

Step 4: Expand the Revenue Equation

By expanding the equation, \[ R(y) = 40000 + 8000y - 2000y - 400y^2 = 40000 + 6000y - 400y^2 \].

Step 5: Identify the Revenue Function Maximum

The revenue function \( R(y) = -400y^2 + 6000y + 40000 \) is a quadratic equation that opens downwards. The maximum occurs at \( y = -\frac{b}{2a} = -\frac{6000}{2*(-400)} = 7.5 \).

Step 6: Calculate the Optimal Room Charge

Substitute \( y = 7.5 \) into the room charge: \( 50 + 10y = 50 + 75 = 125 \).

Key Concepts

Quadratic Functions

At the heart of revenue optimization, particularly in scenarios similar to the Mega Motel chain problem, lies the concept of quadratic functions. Quadratic functions are mathematical expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions graph as a U-shaped curve known as a parabola.

Quadratic functions are crucial in revenue problems due to their ability to model relationships where the effect of a change can both positively and negatively impact the outcome. For instance, when increasing the price of a motel room, it might initially increase revenue, but after a certain point, it will likely cause the revenue to drop due to fewer room bookings.

This is why understanding the shape and properties of a parabola is important: it helps identify the maximum or minimum points that are essential for decision-making in business strategies.

• The coefficient \( a \) determines if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).
• The vertex of the parabola, given by \( -\frac{b}{2a} \), indicates the maximum or minimum point of the function.
• The roots of the function can be found using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which are points where the function intersects the x-axis.

Maximizing Revenue

Maximizing revenue is one of the core objectives in business operations and it often involves carefully balancing price and quantity sold. In the Mega Motel problem, this is explored through the use of a quadratic revenue function. Revenue is calculated as the product of the price per room and the number of rooms booked: \( R = P \times Q \). Here, \( P \) is the adjusted room rate, and \( Q \) is the number of rooms occupied, which decreases as the price increases.

The specific revenue function for the Mega Motel chain was found to be \( R(y) = -400y^2 + 6000y + 40000 \), where \( y \) represents the number of \( 10 \) dollar increases in the room rate. This function forms a downward-opening parabola (since \( a < 0 \)), and its vertex gives the maximum revenue point.

The vertex, calculated by \( y = -\frac{b}{2a} \), tells us the optimal number of price increases needed to maximize revenue. Applying this to the problem, \( y = 7.5 \) means that the optimal room charge, considering the impact on room occupancy, is \( 125 \) per night. This delicate balance ensures the highest possible revenue under the given constraints.

Problem Solving with Equations

Problem solving in mathematics often involves translating real-world situations into equations that can be manipulated to uncover insights. In the context of the Mega Motel problem, this process begins with defining variables and relationships based on the given scenario. We start by expressing the room charge and number of rooms occupied in terms of mathematical variables.

This leads to establishing an equation that represents the key objective—in this case, revenue. After determining the impact of pricing on occupancy, a revenue equation \( R(y) = (50 + 10y)(800 - 40y) \) is formulated. Solving this equation involves steps like expanding and transforming it into a standard quadratic form: \( R(y) = -400y^2 + 6000y + 40000 \).

Finding the maximum revenue points to further problem-solving techniques, such as deriving the vertex of the quadratic function. Identifying this vertex is crucial because it reveals the optimal price increase \( y \) that maximizes revenue. Solving such equations requires a deep understanding of algebraic manipulation, strategic substitutions, and contextual interpretation—not just to solve the problem at hand, but to make informed decisions in similar real-world scenarios.