A company produces a product for which the variable cost is ...

A company produces a product for which the variable cost is $\$ 68.75$ per unit and the fixed costs are $\$ 248,000 .$ The product sells for $\$ 99.99 .$ Let $x$ be the number of units produced and sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost $C$ as a function of the number of units produced. (b) Write the revenue $R$ as a function of the number of units sold. (c) Write the profit $P$ as a function of the number of units sold. (Note: $P=R-C.)$ (d) Use the model in part (c) to find $P(20,000) .$ Interpret your result in the context of the situation. (e) Use the model in part (c) to find $P(0) .$ Interpret your result in the context of the situation.

A company produces a product for which the variable cost is $\$ 68.75$ per unit and the fixed costs are $\$ 248,000 .$ The product sells for $\$ 99.99 .$ Let $x$ be the number of units produced and sold.
(a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost $C$ as a function of the number of units produced.
(b) Write the revenue $R$ as a function of the number of units sold.
(c) Write the profit $P$ as a function of the number of units sold. (Note: $P=R-C.)$
(d) Use the model in part (c) to find $P(20,000) .$ Interpret your result in the context of the situation.
(e) Use the model in part (c) to find $P(0) .$ Interpret your result in the context of the situation.

Key Concepts

Variable Cost and Fixed Cost

These are two fundamental components of a business’s cost structure. Variable costs change in proportion to the number of units produced, while fixed costs remain constant regardless of the production level. Understanding both types of costs is essential for analyzing a company's overall expenses and making production decisions.

Cost Function

The cost function represents the total production cost by combining both variable costs and fixed costs. It is typically expressed as a linear function where the variable part depends on the number of units produced and a constant term represents the fixed cost, enabling businesses to budget and forecast their expenses effectively.

Revenue Function

The revenue function calculates the total income generated from sales by multiplying the number of units sold by the price per unit. It is a key concept in determining the financial performance of a product or service, as it forecasts the income based on sales volume.

Profit Function

The profit function is derived by subtracting the total cost function from the revenue function. This function identifies the net gain or loss from operations, serving as an essential tool for assessing the viability and profitability of a business venture.

Function Evaluation and Interpretation

Evaluating these functions at specific values allows for practical interpretation of business performance. For instance, determining the profit for particular production levels helps in understanding real-world implications, such as how many units must be sold to achieve profitability or what the financial impact is when production is halted.

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