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In the world of business mathematics, the cost function is a fundamental concept used to calculate the total cost associated with producing goods. The cost function consists of two main components: fixed costs and variable costs. Fixed costs are the expenses that do not change with the level of production, such as rent or salaries. These are the costs you incur even if production is zero.

On the other hand, variable costs depend on the volume of production, like the cost of raw materials or direct labor for each unit produced. In our example, the fixed costs are given as \(12,100.00, which are costs not related to the quantity of goods produced. The variable cost is \)0.60 per twin-pack of light bulbs produced.

Thus, the cost function \( C(x) \), which represents the total cost of producing \( x \) twin-packs, is formulated as:

• Fixed Costs: \(12,100.00
• Variable Costs: \)0.60 per twin-pack
• Cost Function Formula: \( C(x) = 12,100 + 0.60x \)

This function provides a complete view of the total cost needed to produce any number of twin-packs and helps in budgeting and financial forecasting.

The revenue function represents the total income generated from selling goods or services. In simpler terms, it's the company's earnings from product sales. For businesses, understanding revenue is crucial as it directly influences profit and financial planning.

The revenue function depends on two factors: the selling price per unit and the number of units sold. In our case, TMI sells each twin-pack of light bulbs for \(1.15. To calculate the revenue, we need to multiply the number of twin-packs sold, \( x \), by the price per twin-pack.

This gives us the revenue function \( R(x) \), which represents the total revenue from selling \( x \) twin-packs:

• Selling Price per Twin-Pack: \)1.15
• Revenue Function Formula: \( R(x) = 1.15x \)

The revenue function is essential for determining how much money the business is bringing in from its sales. By knowing this, businesses can set targets, measure performance, and plan for growth.

The profit function is key in determining a business's financial success, representing the balance between revenue and costs. Simply put, it's the money the company keeps after covering the costs of production. Understanding profit helps businesses to strategize, optimize operations, and make informed decisions about pricing, investment, and expansion.

The profit function is defined as the difference between the revenue function and the cost function. That is, it shows what remains when costs are subtracted from revenue. In our example, the revenue function is \( R(x) = 1.15x \) and the cost function is \( C(x) = 12,100 + 0.60x \).

Thus, we establish the profit function \( P(x) \) as:

• Profit Function Formula: \( P(x) = R(x) - C(x) \)
• Derivative: \( P(x) = 1.15x - (12,100 + 0.60x) \)
• Final Formula: \( P(x) = 0.55x - 12,100 \)

This function is crucial for understanding the underlying profitability of producing and selling a particular number of products. It informs decision-making about whether production levels and pricing strategies are optimal for a business's financial health.