Problem 99 - College Algebra 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: \(C(x)=600,000+45 x \qquad\) This function models the company’s costs. \(R(x)=65 x. \qquad \) 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

The profits for selling 20,000, 30,000 and 40,000 radios respectively are the results from the above calculations. They represent the amount of money the company makes after accounting for production costs.

Step 1: Calculate Profit at x = 20,000

Calculate \( R(20,000) - C(20,000) \). Substitute \( x = 20,000 \) into the revenue function \( R(x) = 65x \) and the cost function \( C(x) = 600,000 + 45x \) to get the revenue and cost respectively, and then subtract the cost from the revenue to get the profit.

Step 2: Calculate Profit at x = 30,000

Calculate \( R(30,000) - C(30,000) \). Substitute \( x = 30,000 \) into the revenue function \( R(x) = 65x \) and the cost function \( C(x) = 600,000 + 45x \) to get the revenue and cost respectively, and then subtract the cost from the revenue to get the profit.

Step 3: Calculate Profit at x = 40,000

Calculate \( R(40,000) - C(40,000) \). Substitute \( x = 40,000 \) into the revenue function \( R(x) = 65x \) and the cost function \( C(x) = 600,000 + 45x \) to get the revenue and cost respectively, and then subtract the cost from the revenue to get the profit.

Step 4: Interpret the Results

The results of the above calculations represent the profit the company makes for selling 20,000, 30,000 and 40,000 radios respectively. A positive value means a profit, while a negative value means a loss.

Key Concepts

Cost Function

The cost function is a mathematical formula that helps us calculate the total cost of producing a certain number of goods or services. In our example, the cost function is given by \[ C(x) = 600,000 + 45x \] where:

• \(600,000\) represents the fixed costs, which we will discuss later;
• \(45x\) represents the variable costs, calculated based on the number of radios produced.

To find out the cost for producing a specific number of radios, simply plug the number into the equation for \(x\). For example, if you want to calculate the cost of producing 20,000 radios, substitute \(x\) with 20,000 in the equation. The result will be the total cost for producing that number of radios.
The cost function is crucial for businesses in determining their overall spending related to production, which further helps in understanding their profit margins.

Revenue Function

The revenue function is a formula used to determine the income generated from selling goods or services. In the context of our example, the revenue function is expressed as \[ R(x) = 65x \] where:

• \(65\) is the price at which each radio is sold;
• \(x\) represents the number of radios sold.

To calculate the revenue for selling a certain number of radios, you must replace \(x\) with the desired number of radios in the formula. This will give you the total revenue generated from those sales.
Understanding the revenue function can help businesses project their sales income, evaluate their pricing strategy, and plan for future growth by showing the potential returns from increasing the number of items sold.

Fixed Costs

Fixed costs are the expenses that do not change, regardless of the quantity of goods or services produced. They are constant and must be paid irrespective of the business's level of production. In the given exercise, the fixed cost is \(600,000\) dollars.
Fixed costs can include:

• Rent for the factory or office space;
• Salaries of permanent employees;
• Utilities;
• Insurance premiums.

These costs are essential for running a business as they represent the basic expenses needed to keep the operation going. By understanding fixed costs, businesses can determine the minimum sales volume required to cover these expenses, which is crucial for maintaining a healthy financial state.

Variable Costs

Variable costs are expenses that change in direct proportion to the level of production activity. Unlike fixed costs, these costs increase as more goods are produced. In the exercise, the variable cost is expressed as \(45x\), indicating that it costs \(45\) dollars to produce each additional radio.
Variable costs might involve:

• Cost of raw materials;
• Costs associated with packaging;
• Wages for temporary or part-time labor;
• Utility costs directly tied to production.

Understanding variable costs is crucial for businesses because it helps in:

• Setting the right price to ensure profitability;
• Making decisions about scaling up production;
• Evaluating efficiency in production processes.

Balancing variable costs with fixed costs allows businesses to better manage their overall expenditure.