Problem 6 - Algebra and Trigonometry

Question

Use the model given to answer the questions about the object or process being modeled. A company finds that the cost \(C\) (in dollars) of manufactur- ing \(x\) compact discs is modeled by $$ C=500+0.35 x $$ Find the cost of manufacturing 1000 compact discs

Step-by-Step Solution

Answer

The cost of manufacturing 1000 compact discs is $850.

Step 1: Understand the Cost Model

The cost model is given by the equation \( C = 500 + 0.35x \), where \( C \) is the total cost in dollars and \( x \) is the number of compact discs produced. The term \( 500 \) represents the fixed costs, and \( 0.35x \) represents the variable costs per compact disc.

Step 2: Plug in the Number of Compact Discs

We need to find the cost of manufacturing 1000 compact discs. Substitute \( x = 1000 \) into the cost equation to find \( C \).

Step 3: Perform the Calculation

Substitute \( x = 1000 \) into the equation:\[ C = 500 + 0.35 \times 1000 \]Calculate the multiplication:\[ 0.35 \times 1000 = 350 \]Now substitute back:\[ C = 500 + 350 \]

Step 4: Simplify to Find the Total Cost

Simplify the equation from the previous step:\[ C = 850 \] This calculation shows that the cost of manufacturing 1000 compact discs is \( 850 \) dollars.

Key Concepts

fixed and variable costs

In any cost model, it's crucial to understand the difference between fixed and variable costs. Fixed costs are constant, regardless of the quantity of items produced. In this exercise, the fixed cost is 500 dollars, which you can think of as expenses that do not change, such as rent or salaries of permanent staff. Fixed costs remain the same no matter how many discs are made.

Variable costs, on the other hand, change depending on the number of items produced. For each compact disc produced, there is an additional cost of 0.35 dollars. This is the variable cost, and it accumulates as production increases. In simpler terms, if you make more discs, expect to pay more in variable costs.

• Fixed costs are unchanging and reflect basic operational expenses.
• Variable costs rise with the level of production.

Understanding these concepts helps businesses predict how production changes affect overall costs.

substitution method

The substitution method is a handy technique in algebra to solve equations by replacing a variable with a given number. In our exercise, we're given a cost model equation: \[ C = 500 + 0.35x \]Here, we are asked to find the cost for manufacturing 1000 compact discs. The substitution method involves taking the value of one variable (in this case, the number of discs, \( x = 1000 \)) and substituting it directly into the equation.

Substitute as follows:

• Replace \( x \) with 1000: \[ C = 500 + 0.35 \times 1000 \]
• Calculate \( 0.35 \times 1000 \), which equals 350.
• Substitute this into the equation: \[ C = 500 + 350 \]

This method is straightforward and gives you the total cost by simply plugging in the values. Use the substitution method whenever you need to evaluate mathematical expressions or models like this.

linear equation in one variable

Linear equations in one variable are equations that involve two terms separated by an equals sign, with one variable. The structure of the equation is typically of the form \( ax + b = c \). Our example is:\[ C = 500 + 0.35x \]This represents a linear relationship between the number of discs produced (\( x \)) and the total manufacturing cost (\( C \)). The goal is usually to find the value of the variable given a specific condition. In this case, we set \( x = 1000 \) to find \( C \).

For any linear equation:

• The graph is a straight line.
• It models direct relationships where one variable changes in proportion to another.

Understanding linear equations in one variable can help solve problems involving proportional relationships, like cost models and budgeting scenarios.