Write a cost function for the problem. Assume that the relationship is linear. Fixed cost, \( \$ 30 ; 5 \) items cost \( \$ 2,970 \) to produce. A. \( C(x)=1,176 x+30 \) B. \( C(x)=588 x+2,970 \) C. \( C(x)=588 x+30 \) D. \( C(x)=1,176 x+2,970 \)

To determine the correct cost function, let's break down the information: 1. The total cost can be expressed as \( C(x) = \text{Fixed Cost} + \text{Variable Cost} \times x \). 2. Given that the fixed cost is \( 30 \), we can write part of the equation as \( C(x) = 30 + \text{Variable Cost} \times x \). 3. The cost for producing 5 items is given as \( 2,970 \). So we can set up the equation: \( 2,970 = 30 + \text{Variable Cost} \times 5 \). 4. Simplifying, \( 2,940 = \text{Variable Cost} \times 5 \), which gives us a variable cost of \( 588 \) per item. Thus, the cost function becomes \( C(x) = 588x + 30 \), making option C the correct choice. Now, let's explore the historical context surrounding linear cost functions. Linear cost functions have their roots in the industrial revolution, where manufacturing processes started to evolve. As factories gained efficiency, understanding costs became paramount, leading to simplified models to predict expenses based on production volume. For real-world application, knowing how to define a cost function is crucial for businesses. Companies can use it to forecast budgets, set prices, or determine break-even points. Understanding your costs helps in making informed decisions that can significantly impact profit margins and overall financial health.