Problem 70 - Thomas Calculus

Question

Three hundred books sell for \(\$ 40\) each, resulting in a revenue of \((300)(\$ 40)=\$ 12,000 .\) For each \(\$ 5\) increase in the price, 25 fewer books are sold. Write the revenue \(R\) as a function of the number \(x\) of \(\$ 5\) increases.

Step-by-Step Solution

Answer

The revenue function is \( R(x) = 12000 + 500x - 125x^2 \).

Step 1: Define Variables

Let \( x \) be the number of \( \\( 5 \) increases in the price per book. The initial price per book is \( \\) 40 \).

Step 2: Create the Expression for New Price

With \( x \) \( \$ 5 \) increases, the new price per book is \( 40 + 5x \).

Step 3: Determine the Number of Books Sold

Initially, 300 books are sold. With each \( \$ 5 \) increase, 25 fewer books are sold, so the number of books sold becomes \( 300 - 25x \).

Step 4: Write the Revenue Function

Revenue \( R \) is given by multiplying the price per book by the number of books sold: \[ R(x) = (40 + 5x)(300 - 25x) \].

Step 5: Simplify the Revenue Function

Expand the expression: \[ R(x) = (40)(300) - (40)(25x) + (5x)(300) - (5x)(25x) \]. Simplifying, we find: \[ R(x) = 12000 + 1500x - 1000x - 125x^2 \], which simplifies to \[ R(x) = 12000 + 500x - 125x^2 \].

Step 6: Finalize the Revenue Function

The function representing revenue \( R \) in terms of \( x \) is: \[ R(x) = 12000 + 500x - 125x^2 \].

Key Concepts

Quadratic Functions

Quadratic functions are polynomials of degree two, which means they have the general form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not equal to zero.
They are known for creating parabolic curves when graphed, which can open upwards or downwards depending on the sign of \( a \). In the case of our revenue function, \( R(x) = 12000 + 500x - 125x^2 \), the quadratic term is \( -125x^2 \).
Since the coefficient of \( x^2 \) is negative, the parabola opens downwards. This means the graph will have a maximum point, which is particularly useful in optimizing tasks.
Expanding quadratic functions involves multiplying out expressions, as shown in the solution with steps like \( (40 + 5x)(300 - 25x) \). Understanding how to expand these correctly is essential because each mathematical operation in the quadratic formula corresponds to a real-world impact on price and number of units sold.

Price Optimization

Price optimization is the process of determining the best price point that maximizes revenue or profit.
In mathematical modeling, especially using quadratic functions, this involves finding the vertex of the parabola, which represents the maximum or minimum point of the function. The vertex can be found using the formula \( x = -\frac{b}{2a} \) for the quadratic function \( ax^2 + bx + c \).
Applying this to our revenue function \( R(x) = 12000 + 500x - 125x^2 \), we calculate the optimal price increase \( x \) by substituting the coefficients into the vertex formula:

• \( a = -125 \)
• \( b = 500 \)

This gives us \( x = -\frac{500}{2(-125)} = 2 \). Thus, the optimal scenario would involve two \( \\(5 \) price increases, leading to a price increase of \( \\)10 \). At this point, the balance between price and quantity results in maximum revenue.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario using equations, functions, and algorithms to predict outcomes or analyze situations.
The revenue function \( R(x) = 12000 + 500x - 125x^2 \) is a product of such modeling, accounting for price changes and variations in demand.The exercise demonstrates how increases in price decrease the number of units sold, aligning with typical supply-demand dynamics seen in economics.
Models like this help businesses understand complex relationships between variables, allowing for informed decision-making.
They employ assumptions (e.g., each \$5 increases results in exactly 25 fewer books sold) and require testing, validation, and refinement to improve accuracy.
A robust model provides valuable insights while illuminating optimal strategies or adjustments needed in operations, pricing, or other factors that can bolster performance in a competitive market. Such models are foundational in analytics and strategy development.