A small software company invests 16,000 to produce a...

A small software company invests $16,000$ to produce a software package that will sell for $855.95 .$ Each unit can be produced for $9.45$ a. Write the cost and revenue functions for $x$ units 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 units that must be sold to break even and verify the result algebraically.

A small software company invests $16,000$ to produce a software
package that will sell for $855.95 .$ Each unit can be produced for $9.45$
a. Write the cost and revenue functions for $x$ units 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 units that must be sold to break even and verify the result algebraically.

Key Concepts

Fixed and Variable Costs

This concept involves distinguishing between costs that do not change with the level of production (fixed costs) and those that vary directly with the number of units produced (variable costs). The fixed cost is incurred regardless of the production volume, while the variable cost increases with each additional unit manufactured.

Cost Function

The cost function represents the total cost of producing a given number of units. It combines both fixed costs and variable costs. Typically, it is expressed as a linear function where the fixed cost is a constant and the variable cost is the product of the cost per unit and the number of units produced.

Revenue Function

The revenue function shows the total income generated from selling a certain number of units. It is calculated by multiplying the sale price per unit by the number of units sold. This function is key in understanding how sales contribute to covering costs and generating profit.

Break-even Analysis

Break-even analysis determines the point at which total revenue equals total costs. At this break-even point, the business neither makes a profit nor incurs a loss. It involves setting the cost function equal to the revenue function and solving for the number of units that must be sold.

Graphical Analysis

Graphical analysis involves plotting the cost and revenue functions on the same axes to visually identify the point of intersection, which represents the break-even point. This method provides an intuitive understanding of how changes in production volume affect both costs and revenues, and it can complement algebraic solutions.

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