Problem 18 - Applied Calculus

Question

A company has cost function \(C(q)=4000+2 q\) dollars and revenue function \(R(q)=10 q\) dollars. (a) What are the fixed costs for the company? (b) What is the marginal cost? (c) What price is the company charging for its product? (d) Graph \(C(q)\) and \(R(q)\) on the same axes and label the break-even point, \(q_{0} .\) Explain how you know the company makes a profit if the quantity produced is greater than \(q_{0}\) (e) Find the break-even point \(q_{0}\).

Step-by-Step Solution

Answer

(a) $4000, (b) $2, (c) $10, (d) Graph shows profit after 500 units, (e) 500 units.

Step 1: Identify Fixed Costs

The fixed costs are the costs that do not change with the quantity produced. These are found in the cost function as the constant term. The given cost function is \(C(q) = 4000 + 2q\), where 4000 is the fixed cost.

Step 2: Determine Marginal Cost

The marginal cost is the cost of producing one more unit of goods. In the cost function \(C(q) = 4000 + 2q\), the marginal cost is represented by the coefficient of \(q\), which is 2.

Step 3: Calculate Price per Unit

The price per unit is determined from the revenue function, \(R(q) = 10q\). Here, the company charges \(10\) dollars per unit because this is the coefficient of \(q\) in the revenue function.

Step 4: Sketch Cost and Revenue Functions

To graph \(C(q)\) and \(R(q)\), plot both functions on the same set of axes. The cost function is a line \(C(q) = 4000 + 2q\) and the revenue function is a line \(R(q) = 10q\). The break-even point is where these two lines intersect. This is the point \(q_{0}\) where the company changes from a loss to a profit.

Step 5: Solve for Break-Even Point

Set the cost function equal to the revenue function to find the break-even point: \(4000 + 2q = 10q\). Solve for \(q\):\[(4000 + 2q = 10q) \4000 = 8q \q = \frac{4000}{8} \q = 500\]Thus, \(q_{0} = 500\) units.

Step 6: Analyze Profitability Beyond Break-Even

The company makes a profit if \(q > q_{0}\). After the break-even point (\(q_{0} = 500\)), the revenue \(R(q)\) exceeds the cost \(C(q)\), indicating that each additional unit sold generates profit.

Key Concepts

Understanding Fixed Costs

Fixed costs are those expenses that a company pays regardless of its production levels. These costs consist of overheads like rent, salaries, and other basic operational expenses that do not vary with the amount produced. In a cost function, fixed costs appear as a constant term because they remain unchanged no matter how much the company produces.
In the example given, the cost function is expressed as:

• \( C(q) = 4000 + 2q \)

Here, the fixed cost is \(4000\) dollars. It's a crucial part of understanding the financial stability of the company because these costs must be paid even if the production is zero. Therefore, understanding and managing fixed costs is vital for any business to ensure sound financial health.

Exploring Marginal Cost

Marginal cost refers to the additional cost a company incurs to produce one extra unit of product. This is a crucial concept because it helps in determining the profitability of producing additional units.
In the formula used for the cost function:

• \( C(q) = 4000 + 2q \)

the marginal cost is \(2\) dollars. It represents the variable part of the cost—the price of increased production. By analyzing the marginal cost, businesses can make informed decisions about scaling production and maximizing profits.

Understanding the Revenue Function

The revenue function illustrates how much money a company earns from selling its products. This is a linear function where the total revenue changes proportionally with sales volume. The coefficient of the variable in the revenue function gives us the price charged for one unit of the product.
In the exercise, the revenue function is:

• \( R(q) = 10q \)

This indicates the company charges \(10\) dollars per unit. Revenue functions are vital for determining sales strategies and understanding how different levels of production and pricing impact the company's income.

Breaking Down the Cost Function

A cost function describes the total cost incurred by a company as a function of the quantity of goods produced. It combines both fixed and variable costs, giving an overall picture of the financial obligations related to production.
In this exercise, the cost function is:

• \( C(q) = 4000 + 2q \)

This equation shows both fixed costs (\(4000\) dollars) and variable costs (\(2q\) dollars). It serves various purposes, like setting pricing strategies by determining the break-even point, which is the production level where total costs equal total revenues. Understanding the cost function is essential to effectively manage resources and optimize production for profitability.