Problem 54 - Beginning and Intermediate Algebra

Question

In each problem, find the following. (a) A function \(R(x)\) that describes the total revenue received (b) The graph of the function from part ( \(a\) ) (c) The number of unsold seats that will produce the maximum revenue (d) The maximum revenue A charter bus charges a fare of \(\$ 48\) per person, plus \(\$ 2\) per person for each unsold seat on the bus. The bus has 42 seats. Let \(x\) represent the number of unsold seats.

Step-by-Step Solution

Answer

Maximum revenue occurs with 9 unsold seats, generating $2178.

Step 1: Define the Revenue Function R(x)

The revenue function is defined by considering the fare charged and the extra charge per unsold seat. Let’s denote the number of unsold seats by \( x \). The bus has a total of 42 seats, so the number of sold seats is \( 42 - x \). The fare charged per person is \( 48 + 2x \). Therefore, the revenue function \( R(x) \) is given by:\[ R(x) = (42 - x)(48 + 2x) \]

Step 2: Simplify the Revenue Function

Simplify \( R(x) \) by expanding the product:\[ R(x) = 42 \cdot 48 + 42 \cdot 2x - x \cdot 48 - x \cdot 2x \]Simplify further:\[ R(x) = 2016 + 84x - 48x - 2x^2 \]Combine like terms:\[ R(x) = 2016 + 36x - 2x^2 \]Rearrange to match standard quadratic form:\[ R(x) = -2x^2 + 36x + 2016 \]

Step 3: Graph the Revenue Function

Plot the quadratic equation \( R(x) = -2x^2 + 36x + 2016 \). This can be done using graphing software or by hand. Identify the vertex of the parabola, which represents the highest point since the parabola opens downwards.

Step 4: Find the Number of Unsold Seats for Maximum Revenue

The maximum revenue occurs at the vertex of the quadratic function. For a quadratic function \( ax^2 + bx + c \), the vertex occurs at \[ x = -\frac{b}{2a} \]. Here \( a = -2 \) and \( b = 36 \), so:\[ x = -\frac{36}{2(-2)} = 9 \]Thus, 9 unsold seats will produce the maximum revenue.

Step 5: Calculate the Maximum Revenue

Substitute \( x = 9 \) into the revenue function \( R(x) = -2x^2 + 36x + 2016 \) to find the maximum revenue:\[ R(9) = -2(9)^2 + 36(9) + 2016 \]Simplify:\[ R(9) = -2 \cdot 81 + 36 \cdot 9 + 2016 \]\[ R(9) = -162 + 324 + 2016 \]\[ R(9) = 2178 \]Therefore, the maximum revenue is \ 2178 \ dollars.

Key Concepts

revenue function

Understanding how to calculate the total revenue received helps in making better business decisions. In our problem, the revenue function is like a recipe that mixes different ingredients: the number of sold seats, the fare per person, and the extra charge for unsold seats.

First, define the variables:
\begin{itemize}

Total seats: 42

Unsold seats: \( x \)

Fare per person: \( 48 + 2x \)

Number of sold seats: \( 42 - x \)

Combining these, the revenue function \( R(x) \) is:
\[ R(x) = (42 - x) (48 + 2x) \] This function will tell you your total revenue based on the number of unsold seats. Simplifying it gives:
\[ R(x) = -2x^2 + 36x + 2016 \] This quadratic equation forms the foundation for further analysis.

parabolic graph

The graph of a quadratic function is always a parabola, a U-shaped curve. In our case, the equation is:
\[ R(x) = -2x^2 + 36x + 2016 \] Since the leading coefficient (the number in front of \( x^2 \)) is negative (-2), the parabola opens downwards.

Key points to graph:

• Intercepts: Start by finding where the graph crosses the y-axis and x-axis.
• Vertex: The highest point on the graph because the parabola opens downward.

Use graphing tools or do it by hand for a visual understanding. The shape of this parabola helps visually identify how revenue changes with different numbers of unsold seats.

vertex of a parabola

The vertex of a parabola is crucial when dealing with quadratic functions. It represents either the maximum or minimum point of the graph. Since our revenue function forms a downward-opening parabola, the vertex gives the maximum revenue.

For any quadratic equation in the form \( ax^2 + bx + c \), the vertex occurs at:
\[ x = -\frac{b}{2a} \] In our problem, \( a = -2 \) and \( b = 36 \), so:
\[ x = -\frac{36}{2 \times -2} = 9 \] This means that having 9 unsold seats will produce the maximum revenue. Find this point in your graph to double-check your work.

maximizing revenue

Knowing the vertex gives the number of unsold seats for maximum revenue. Plug this value back into your revenue function to find the actual dollar amount.

With \( x = 9 \) in our simplified function \( R(x) = -2x^2 + 36x + 2016 \):
\[ R(9) = -2(9)^2 + 36(9) + 2016 \] Simplifying:
\[ R(9) = -2 \times 81 + 36 \times 9 + 2016 \] \[ R(9) = -162 + 324 + 2016 = 2178 \] So the maximum revenue obtainable is \$ 2178. Remember, the steps you follow allow you to apply these techniques to various situations where maximizing revenue is essential. Break down each element, and you'll steer your way to the highest earnings.