Break-Even Analysis A small fast food restaurant invests ...

Break-Even Analysis A small fast food restaurant invests $\$ 5000$ to produce a new food item that will sell for $\$ 3.49$ . Each item can be produced for $\$ 2.16$ . (a) Write the cost and revenue functions for $x$ items produced and sold. (b) Use a graphing utility to graph the cost and revenue functions in the same viewing window. Use the graph to approximate the number of items that must be sold to break even, and verify the result algebraically.

Break-Even Analysis A small fast food restaurant invests $\$ 5000$ to produce a new food item that will sell for $\$ 3.49$ . Each item can be produced for $\$ 2.16$ .
(a) Write the cost and revenue functions for $x$ items produced and sold.
(b) Use a graphing utility to graph the cost and revenue functions in the same viewing window. Use the graph to approximate the number of items that must be sold to break even, and verify the result algebraically.

Key Concepts

Graphical Analysis

Graphical analysis involves plotting the cost and revenue functions on a coordinate plane to visually identify their intersection point, which represents the break-even point. This method provides an intuitive understanding of how costs and revenues compare across different levels of production and assists in decision making.

Cost Function

A cost function represents the total cost incurred in producing goods, which includes both fixed costs (costs that do not change with the quantity produced) and variable costs (costs that vary depending on the production volume). This function is typically expressed as a linear equation where the fixed cost is added to the product of the variable cost and the quantity produced.

Revenue Function

A revenue function calculates the total income generated from selling goods, and is usually defined as the product of the selling price per unit and the number of units sold. This function is key for determining how much income is generated at different levels of production and sales.

Fixed and Variable Costs

Fixed costs are expenditures that remain constant regardless of the production volume, such as initial investments or rent, while variable costs change in proportion to the number of items produced. Understanding the distinction between these two is crucial for constructing accurate cost functions and analyzing profitability.

Break-even Point

The break-even point is the production quantity at which total revenue equals total cost, resulting in zero profit. Calculating this point helps businesses understand the minimum number of units that need to be sold to cover all costs, and is a fundamental concept in cost analysis and financial planning.

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