Problem 58 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A manufacturer of tennis rackets finds that the total cost of manufacturing \(x\) rackets/day is given by $$ 0.0001 x^{2}+4 x+400 $$ dollars. Each racket can be sold at a price of \(p\) dollars, where $$ p=-0.0004 x+10 $$ Find an expression giving the daily profit for the manufacturer, assuming that all the rackets manufactured can be sold.

Step-by-Step Solution

Answer

The daily profit for the manufacturer is given by the expression \(P(x) = -0.0005x^2 + 6x - 400\).

Step 1: Find the revenue function

To find the revenue function, we need to multiply the number of rackets (\(x\)) with the price function. The price function is given as \(p = -0.0004x + 10\). Therefore, the revenue function is: Revenue (R) = Price × Number of rackets = \(p \times x = x(-0.0004x + 10)\)

Step 2: Simplify the revenue function

Now, let's simplify the revenue function by multiplying and combining like terms: R = \((-0.0004x^2 + 10x)\)

Step 3: Find the profit function

The profit function is the difference between the revenue function and the total cost function. We already have the revenue function as R = \((-0.0004x^2 + 10x)\) and the total cost function is given as \(C = 0.0001x^2 + 4x + 400\). Subtracting the cost function from the revenue function, we have: Profit (P) = Revenue - Cost = \((-0.0004x^2 + 10x) - (0.0001x^2 + 4x + 400)\)

Step 4: Simplify the profit function

Now, let's simplify the profit function by combining like terms: P = \((-0.0004x^2 + 10x) - (0.0001x^2 + 4x + 400) = -0.0004x^2 + 10x - 0.0001x^2 - 4x - 400\) P = \((-0.0004x^2 - 0.0001x^2) + (10x - 4x) - 400 = -0.0005x^2 + 6x - 400\) So, the expression for the daily profit for the manufacturer is given by: \(P(x) = -0.0005x^2 + 6x - 400\).

Key Concepts

Revenue Function

When it comes to understanding business concepts in mathematics, the revenue function plays a critical role. It is essentially a mathematical expression that calculates the total revenue generated by a company from selling a certain number of products or services.

In the exercise we're considering, the manufacturer earns revenue by selling tennis rackets. The price for each racket isn't fixed but changes depending on how many rackets are sold in a day. This dynamic pricing is represented by the price function:

• Price, \( p = -0.0004x + 10 \), where \( x \) is the number of rackets sold.

The revenue, therefore, can be found by multiplying this variable price by the number of rackets sold (

• Revenue function, \( R(x) = p \times x \), becomes \( R(x) = x(-0.0004x + 10) \).

These functions allow businesses to model income, making them indispensable tools for financial forecasting and decision-making.

Cost Function

Cost functions are just as crucial as revenue functions for businesses, as they capture all expenses incurred in the production process. In our example, the cost function offers insight into how much it costs to manufacture a certain number of tennis rackets per day.

• Total cost function, \( C(x) = 0.0001x^{2} + 4x + 400 \), where \( x \) stands for the number of rackets.

This function includes both fixed costs, which do not change with the number of units produced (like the cost of maintaining factory machinery), and variable costs, which change depending on production volume (like raw materials).

By using the cost function, the company can predict how its spending will change as production levels increase or decrease.

Quadratic Functions

Quadratic functions are central to many types of real-world modeling, including financial modeling. Both the revenue and cost functions in the tennis racket manufacturer exercise are quadratic functions. A quadratic function takes the general form

• \( ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In the context of this problem:

• The revenue function is \(-0.0004x^2 + 10x\), and the cost function is \(0.0001x^2 + 4x + 400\).

Quadratic equations graph to form parabolas, which have a characteristic "U" or inverted "U" shape. Understanding these shapes helps in analyzing and predicting trends, such as identifying maximum profit points.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world scenarios. It offers a simplified representation of a complex process and is vital in decision-making. In the exercise provided, we use mathematical models to address a business challenge.

• The **profit function**, derived from subtracting the cost function from the revenue function, helps predict financial outcomes: \( P(x) = -0.0005x^2 + 6x - 400\).

Through these models, the manufacturer can predict how changes in production levels affect profits. It enables strategic planning like assessing the economic impact of producing more or fewer units.

Ultimately, mathematical modeling allows decision-makers to visualize possible financial outcomes and make informed choices.