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

A company that sells radios has yearly fixed costs of 600,000 dollar. It costs the company 45 dollar to produce each radio. Each radio will sell for 65 dollar. The company's costs and revenue are modeled by the following functions: $C(x)=600,000+45 x$ $R(x)=65 x$ 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 dollar. It costs the company 45 dollar to produce each radio. Each radio will sell for 65 dollar. The company's costs and revenue are modeled by the following functions:
$C(x)=600,000+45 x$
$R(x)=65 x$
Find and interpret $(R-C)(20,000),(R-C)(30,000),$ and $(R-C)(40,000)$

Key Concepts

Profit Function

The profit function is obtained by subtracting the cost function from the revenue function. It measures the net gain or loss, and in this problem is calculated as R(x) - C(x) = 65x - (600,000 + 45x) = 20x - 600,000.

Interpretation of Profit Values

Evaluating the profit function at different production levels, such as 20,000, 30,000, and 40,000 radios, provides insight into the company’s profitability at those levels. A negative result indicates a loss, while a positive result signifies profit, and it helps determine the scale at which the company breaks even or becomes profitable.

Revenue Function

The revenue function represents the income from selling products. In this scenario, each radio is sold for $65, leading to a revenue function R(x) = 65x, where x is the number sold.

Fixed Costs

Fixed costs are expenses that remain constant regardless of how much is produced. In this context, the company’s yearly cost of $600,000 is incurred whether a few or many radios are produced.

Variable Costs

Variable costs change in direct proportion to the number of units produced. Here, the cost of $45 per radio is a variable cost, which when multiplied by the number of radios produced, gives the total variable cost.

Cost Function

The cost function represents total expenses by adding fixed costs and variable costs. It is expressed as C(x) = 600,000 + 45x, where x is the number of radios produced.

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