Problem 48 - Essential Calculus Early Transcendentals

Question

The manager of a 100 -unit apartment complex knows from experience that all units will be occupied if the rent is \(\$ 800\) per month. A market survey suggests that, on average, one additional unit will remain vacant for each \(\$ 10\) increase in rent. What rent should the manager charge to maximize revenue?

Step-by-Step Solution

Answer

The optimal rent to maximize revenue is $900 per month.

Step 1: Understand the Problem

We have a 100-unit apartment complex. All units are occupied if the rent is $800/month. For every $10 increase in rent, one more unit becomes vacant.

Step 2: Define Variables

Let \( x \) be the number of $10 increases in rent. The rent for each unit becomes \( 800 + 10x \). The number of occupied units becomes \( 100 - x \).

Step 3: Formulate the Revenue Function

Revenue \( R \) is the product of rent per unit and the number of occupied units. Therefore, \( R(x) = (800 + 10x)(100 - x) \).

Step 4: Expand the Revenue Function

Expand the equation: \( R(x) = (800 + 10x) \cdot 100 - (800 + 10x) \cdot x \), which simplifies to \( R(x) = 80000 + 1000x - 800x - 10x^2 \).

Step 5: Simplify the Revenue Function

Combine like terms to get: \( R(x) = 80000 + 200x - 10x^2 \).

Step 6: Find the Maximum Revenue

This is a quadratic function \( R(x) = -10x^2 + 200x + 80000 \). The maximum revenue occurs at the vertex of the parabola. Use \( x = -\frac{b}{2a} \), where \( a = -10 \) and \( b = 200 \).

Step 7: Calculate the Vertex

Substitute into the vertex formula: \( x = -\frac{200}{2(-10)} = 10 \).

Step 8: Determine Optimal Rent Value

Substitute \( x = 10 \) back into the rent formula: rent = \( 800 + 10x = 800 + 100 = 900 \).

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function that is represented by the general equation \( ax^2 + bx + c \). It forms a parabola on the graph. The parabola can open upwards or downwards, depending on the coefficient \( a \). If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards.

In the context of the problem, the function \( R(x) = -10x^2 + 200x + 80000 \) is a quadratic function. Here, \( a = -10 \), \( b = 200 \), and \( c = 80000 \). The negative \( a \) value indicates that the parabola opens downwards.

Understanding quadratic functions is crucial because many real-world optimization problems can be modeled using them. The maximum or minimum point of the parabola (its vertex) helps in finding optimal solutions, like maximizing revenue in our exercise.

Revenue Maximization

Revenue maximization in this setting means finding the optimal rent price that results in the highest possible income for the apartment complex.

To do this, we first need to define a revenue function. This function \( R(x) \) represents the total revenue generated and is calculated as the product of the number of units rented and the rent price per unit. As shown in the exercise, the function is \( R(x) = (800 + 10x)(100 - x) \).

This formula is then expanded and simplified to form a quadratic equation, \( R(x) = -10x^2 + 200x + 80000 \). By identifying the vertex of this parabola, we find the value of \( x \) that maximizes revenue. This optimization helps businesses like apartment managers decide the best pricing strategies for maximizing profits.

Vertex Formula

The vertex formula is used to find the highest or lowest point, known as the vertex, on a parabola described by a quadratic equation. The formula is \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are coefficients from the quadratic equation \( ax^2 + bx + c \).

In the current situation, we apply this formula to the revenue function \( R(x) = -10x^2 + 200x + 80000 \). Plugging in the values, we find \( x = -\frac{200}{2(-10)} = 10 \).

This means the vertex occurs when \( x = 10 \). Using this \( x \), we can find the rent that maximizes revenue by substituting \( x = 10 \) back into the rent formula: \( 800 + 10x = 900 \). The vertex formula thus provides a simple yet powerful method for identifying optimization points in quadratic functions.