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 $\$ 55.95 .$ 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 $\$ 55.95 .$ 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

Cost Function

The cost function represents the total cost incurred when producing a certain number of units. It is typically composed of a fixed cost, which remains constant regardless of the number of units produced, and a variable cost, which increases with each additional unit produced. This function helps in budgeting and planning for production expenses.

Revenue Function

The revenue function represents the total income generated from selling a certain number of units at a given price per unit. It is calculated as the product of the number of units sold and the price per unit. This function is essential for forecasting sales and determining profitability.

Break-even Analysis

Break-even analysis involves finding the point where the total revenue equals the total cost. At this point, the business neither makes a profit nor incurs a loss. Determining the break-even quantity is crucial for understanding the minimum sales needed to cover all expenses and for making informed decisions about production and pricing.

Graphical Analysis of Functions

Graphical analysis involves plotting functions, such as cost and revenue functions, on the same coordinate system to visually determine their point of intersection. The intersection point represents the break-even point where revenue equals cost. This method provides an intuitive way to validate algebraic solutions and to understand the relationship between different economic factors.

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