Problem 99 - Precalculus Essentials

Question

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: \(C(x)=600,000+45 x\) This function models the company's costs. \(R(x)=65 x\) This function models the company's revenue. Find and interpret \((R-C)(20,000),(R-C)(30,000),\) and \((R-C)(40,000)\)

Step-by-Step Solution

Answer

For 20,000 radios, the company will have a loss of \$200,000. For 30,000 radios, the company will neither profit nor lose, as they will just break even. For 40,000 radios, the company will have a profit of \$200,000.

Step 1: Understand the given functions

The costs function and revenue function are given by \(C(x) = 600,000 + 45x\) and \(R(x) = 65x\) respectively. Where 'x' is the number of radios produced and sold. \(C(x)\) models the company's total costs (fixed costs + production costs). \(R(x)\) models the company's revenue (number of radios multiplied by the selling price per radio).

Step 2: Calculate (R-C)(20,000)

First, substitute \(x = 20,000\) into both functions so that we calculate the cost and revenue for 20,000 radios, then subtract the total cost from the total revenue to find the profit or loss. So \[(R-C)(20,000) = R(20,000) - C(20,000) = (65 * 20,000) - (600,000 + 45 * 20,000) = 1,300,000 - 1,500,000 = -200,000.\]

Step 3: Calculate (R-C)(30,000)

Now, substitute \(x = 30,000\) into both functions, then subtract total cost from total revenue. So \[(R-C)(30,000) = R(30,000) - C(30,000) = (65 * 30,000) - (600,000 + 45 * 30,000) = 1,950,000 - 1,950,000 = 0.\]

Step 4: Calculate (R-C)(40,000)

Finally, substitute \(x = 40,000\) into both functions, then subtract total cost from total revenue. So \[(R-C)(40,000) = R(40,000) - C(40,000) = (65 * 40,000) - (600,000 + 45 * 40,000) = 2,600,000 - 2,400,000 = 200,000.\]

Key Concepts

Cost Function

The cost function, denoted as \(C(x)\), in business terms, represents the total expenses that a company incurs in producing its goods. It usually combines two main components:

• The fixed costs
• The variable costs per unit

In the case of the radio-selling company, the cost function given is:\[C(x) = 600,000 + 45x\]Here, the fixed cost is \(\\(600,000\) and does not change regardless of the number of radios produced. The variable cost is \(\\)45\) per radio, which means for each additional radio produced, the total cost increases by \(\$45\). The variable part depends directly on the number of radios produced, represented by \(x\).
The cost function allows the business to predict total production costs by inputting any value for \(x\). With this, the company can determine the expenses associated with producing a certain number of products.

Revenue Function

The revenue function, expressed as \(R(x)\), describes the total income a company expects from sales before deducting any costs. It is calculated by multiplying the number of units sold by the price per unit. For the radio company, the function is:\[R(x) = 65x\]This expression shows that the revenue comes from selling radios at \(\$65\) each. The total revenue increases with each radio sold, directly proportional to \(x\), the number of radios. In essence, revenue functions give insight into potential earnings and help businesses project how sales volume impacts total income.
By comparing this function with cost functions, businesses can quickly calculate profits or losses by subtracting total costs from revenue using different quantities of \(x\). It's a crucial part of financial analysis for setting sales targets and strategic planning.

Fixed Costs

Fixed costs refer to the expenses that do not vary with the level of production or sales. They remain constant irrespective of whether the company sells no products or a million. In our exercise, the fixed cost is stated as \(\$600,000\).
Examples of fixed costs typically include:

• Rent or mortgage payments
• Salaries of permanent staff
• Insurance premiums
• Loan repayments

These are critical in understanding how certain expenses are essential for a business's operation, independent of how many products it sells. Knowing fixed costs helps companies determine break-even points and understand the minimum revenue needed to cover these expenses. It also assists in long-term financial planning, as these costs provide a baseline of financial commitment irrespective of business performance.

Variable Costs

Variable costs are the expenses that change directly in proportion to the level of production output. The more a company produces, the higher these costs become. For the radio company, every radio produced increases the cost by \(\$45\), which is tracked by the term \(45x\).
Typical examples of variable costs include:

• Raw materials
• Production supplies
• Direct labor costs (if paid per unit produced)
• Shipping costs

Understanding variable costs allows businesses to estimate how expenses will grow as sales increase and helps in establishing pricing strategies to ensure profitability. Unlike fixed costs, managing and optimizing variable expenses often provide companies greater flexibility in adapting to changing market conditions, influencing competitiveness and profitability.