Problem 109 - Calculus Volume 1

Question

The total cost \(C\) (in thousands of dollars) to produce a certain item is modeled by the function \(C(x)=10.50 x+28,500,\) where \(x\) is the number of items produced. Determine the cost to produce 175 items.

Step-by-Step Solution

Answer

The cost to produce 175 items is $30,337.50, in thousands of dollars.

Step 1: Understand the Function

The function given is \( C(x) = 10.50x + 28,500 \). This represents the total cost \( C \) in thousands of dollars as a linear function of \( x \), the number of items produced. The term \( 10.50x \) represents the variable cost per item, and \( 28,500 \) is the fixed cost.

Step 2: Substitute the Number of Items Produced

To find the cost of producing 175 items, substitute \( x = 175 \) into the function: \( C(175) = 10.50 \times 175 + 28,500 \).

Step 3: Calculate the Variable Cost

Calculate the variable cost, which is \( 10.50 \times 175 \). This simplifies to \( 1,837.50 \).

Step 4: Add the Fixed Cost

Add the fixed cost \( 28,500 \) to the calculated variable cost to find the total cost: \( C(175) = 1,837.50 + 28,500 \).

Step 5: Calculate the Total Cost

The total cost, \( C(175) \), is calculated as \( 1,837.50 + 28,500 = 30,337.50 \). This represents the cost in thousands of dollars.

Key Concepts

Cost Calculation

Cost calculation is a fundamental concept in finance and business. It enables companies to estimate the amount needed to produce goods or services. In our example, the cost function gives us a mathematical model that includes two parts: fixed costs and variable costs. To calculate the total cost, you need to consider both these elements, represented by the equation \( C(x) = 10.50x + 28,500 \). Here, \( C(x) \) is the total cost, \( x \) is the quantity of items, and both costs are included:

• Variable costs (\(10.50x\)) are tied to the production level. More production means higher variable costs.
• Fixed costs (\(28,500\)) remain constant regardless of production level.

By substituting the number of items into the function, you can easily compute the corresponding cost.

Fixed and Variable Costs

Fixed and variable costs are crucial in distinguishing how various costs behave in response to changes in production levels. **Fixed costs** do not change with the number of items produced. They might include rent, salaries, or equipment that a company must pay regardless of production. In our function, this is represented by \( 28,500 \), a constant term.
**Variable costs**, on the other hand, increase or decrease according to how much a company produces. They are directly correlated with output, so producing more items means greater variable costs. In the given function, \( 10.50x \) indicates the cost per additional item.

• Understanding these cost types helps in budgeting and pricing strategies.
• Proper analysis of fixed and variable costs is essential for financial planning and forecasting.

Recognize how each component affects the overall cost calculation to optimize production efficiency.

Substitution in Functions

Substitution in functions is a straightforward method used to determine specific outcomes by replacing a variable with a given number. In cost calculations, this technique helps you find the total cost for a certain number of items. Given the function \( C(x) = 10.50x + 28,500 \), substituting \( x = 175 \) allows us to compute the cost for producing 175 items.
This involves replacing \( x \) in the formula with the desired value and performing the necessary arithmetic operations. Here are the steps:

• Substitute \( x \) with 175: \( C(175) = 10.50 \times 175 + 28,500 \).
• Calculate the variable cost by multiplying 175 by 10.50, which results in 1,837.50.
• Add the fixed cost: 1,837.50 + 28,500 = 30,337.50.

By performing substitution, you find the specific total cost, making it a valuable tool in various applications beyond this example.