For each of the following pairs of total-cost and total...

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point.
$$
\begin{array}{l}
{C(x)=20 x+120,000} \\
{R(x)=50 x}
\end{array}
$$

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Key Concepts

Break-even Point

The break-even point is the level of output at which total revenue exactly equals total cost, resulting in zero profit. Determining this point is crucial as it helps businesses understand the minimum sales volume needed to cover all expenses, thereby guiding pricing and production strategies.

Total Cost Function

This function represents the overall cost incurred in the production or provision of goods and services, incorporating both fixed costs (costs that remain constant regardless of the quantity produced) and variable costs (costs that change with the level of output). Understanding the total cost function is essential for analyzing production expenses and determining profitability.

Total Revenue Function

This function calculates the total earnings a firm generates from selling its products or services, typically represented as the product of the price per unit and the number of units sold. It forms the basis for comparing earnings against costs to assess business performance.

Profit Function

The profit function is derived by subtracting the total cost function from the total revenue function. It quantitatively expresses the company's earnings (or losses) based on different levels of production, making it a fundamental tool in financial analysis and decision-making.

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