Problem 7 - Algebra and Trigonometry

Question

Use the model given to answer the questions about the object or process being modeled. A company models the profit \(P(\text { in dollars) on the sale of }\) \(x\) CDs by $$ P=0.8 x-500 $$ Find the profit on the sale of 1000 CDs.

Step-by-Step Solution

Answer

The profit on the sale of 1000 CDs is $300.

Step 1: Identify the Given Information

We are given the profit function as \( P = 0.8x - 500 \), where \( x \) represents the number of CDs sold. We need to find the profit when 1000 CDs are sold.

Step 2: Substitute the Number of CDs into the Profit Function

Substitute \( x = 1000 \) into the profit function: \[ P = 0.8(1000) - 500 \]

Step 3: Calculate the Product

Calculate the product \( 0.8 \times 1000 \):\[ 0.8 \times 1000 = 800 \]

Step 4: Subtract to Find the Profit

Subtract 500 from 800 to find the profit:\[ P = 800 - 500 = 300 \]

Step 5: Conclusion on Profit Calculation

The profit from selling 1000 CDs is 300 dollars. We have used the function to substitute the value of \( x \) and found the net profit.

Key Concepts

Linear Modeling

Linear modeling plays a crucial role when predicting or analyzing trends. In the context of business, it helps in estimating profits, costs, and other financial metrics. A linear model is generally represented in the form of a linear equation, where the relationship between variables is depicted through a straight line. In our provided exercise, the model is summarized in the equation \( P = 0.8x - 500 \). This equation reveals how profit \( P \) varies with the number of CDs sold \( x \).
\( P \) and \( x \) are related linearly, meaning for each additional CD sold, the profit increases uniformly by a fixed amount, specifically 0.8 dollars in this case. The term \( -500 \) represents a constant deduction from the total sales. This could include initial costs or a fixed overhead that the company must cover before making a profit. To utilize the model effectively, understanding how each component contributes to the overall profit calculation is essential.
By using linear modeling, companies can make informed decisions about production levels and price settings, ensuring alignment with their financial goals.

Substitution Method

The substitution method is a straightforward technique often used in algebra to find specific values within an equation. This method involves replacing a variable with a given value to simplify the problem and find a solution. In our profit function \( P = 0.8x - 500 \), we use substitution to determine what the profit would be when a particular number of CDs, like 1000, is sold.
To apply this method, follow these simple steps:

• Identify the variable that needs substitution. In our case, it's \( x \), the number of CDs.
• Replace \( x \) with the specific number given, which is 1000 in this instance.
• Simplify the equation by carrying out the arithmetic operations.

The advantage of using the substitution method is that it provides a clear, direct way to work with equations, especially when dealing with linear models. It allows you to translate theoretical models into practical financial insights with ease.

Algebraic Calculation

Algebraic calculation is at the heart of solving equations once variables are substituted. It involves performing arithmetic operations such as multiplication, division, addition, and subtraction. In our example, after substituting \( x = 1000 \) into the equation \( P = 0.8x - 500 \), we proceed with algebraic calculations.
The first step is multiplication: calculate \( 0.8 \times 1000 \). This results in 800, which represents the total revenue from selling 1000 CDs before deducting fixed costs. Next, subtract the fixed cost represented by 500 from this amount of 800. This gives you the final profit: \( P = 800 - 500 = 300 \).
The precision of algebraic calculations is essential, as errors could lead to incorrect financial analyses. By accurately performing these calculations, businesses can reliably predict profits and costs, allowing for better financial planning and decision-making.