A company that sells radios has yearly fixed costs of ...

A company that sells radios has yearly fixed costs of $\$ 600,000 .$ It costs the company $\$ 45$ to produce each radio. Each radio will sell for $\$ 65 .$ The company's costs and revenue are modeled by the following functions, where $x$ represents the number of radios produced and sold: $$\begin{array}{l}{C(x)=600,000+45 x} \\ {R(x)=65 x}\end{array}$$ Find and interpret $(R-C)(20,000),(R-C)(30,000),$ and $(R-C)(40,000).$

A company that sells radios has yearly fixed costs of $\$ 600,000 .$ It costs the company $\$ 45$ to produce each radio. Each radio will sell for $\$ 65 .$ The company's costs and revenue are modeled by the following functions, where $x$ represents the number of radios produced and sold:
$$\begin{array}{l}{C(x)=600,000+45 x} \\ {R(x)=65 x}\end{array}$$
Find and interpret $(R-C)(20,000),(R-C)(30,000),$ and $(R-C)(40,000).$

Key Concepts

Interpretation of Profit Values

Interpreting the profit values obtained through evaluation involves assessing whether a given production level leads to a profit or a loss. Positive profit values indicate that revenues exceed costs at that level, while negative values suggest that costs surpass revenues.

Function Evaluation

Function evaluation involves substituting specific values (in this context, quantities of product sold and produced) into the cost, revenue, or profit functions. This process provides concrete numerical outcomes that are critical for financial decision-making and analysis.

Revenue Function

The revenue function calculates the total income generated from selling a product. It is usually modeled as the product of the unit selling price and the number of units sold, reflecting how revenue scales with sales volume.

Cost Function

The cost function represents the total expenses incurred by a company in producing goods. It typically consists of fixed costs, which remain constant regardless of the output level, and variable costs, which change in proportion to the quantity produced.

Profit Function

The profit function is derived from subtracting the total costs from the total revenue. It represents the net gain or loss, and analyzing it helps determine the profitability of producing and selling different quantities of a product.

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