Problem 8 - Applied Calculus

Question

A company has cost and revenue functions, in dollars, given by \(C(q)=6000+10 q\) and \(R(q)=12 q\) (a) Find the cost and revenue if the company produces 500 units. Does the company make a profit? What about 5000 units? (b) Find the break-even point and illustrate it graphically.

Step-by-Step Solution

Answer

500 units: loss of $5,000; 5000 units: profit of $4,000. Break-even point at 3000 units.

Step 1: Evaluate Cost and Revenue for 500 Units

To find the cost for producing 500 units, substitute \( q = 500 \) in the cost function: \( C(500) = 6000 + 10 \times 500 = 6000 + 5000 = 11000 \). For revenue, substitute in the revenue function: \( R(500) = 12 \times 500 = 6000 \). Therefore, at 500 units: Cost = \(11,000, Revenue = \)6,000, resulting in a loss of $5,000.

Step 2: Evaluate Cost and Revenue for 5000 Units

Calculate the cost for 5000 units: \( C(5000) = 6000 + 10 \times 5000 = 6000 + 50000 = 56000 \). For revenue: \( R(5000) = 12 \times 5000 = 60000 \). Therefore, at 5000 units: Cost = \(56,000, Revenue = \)60,000, resulting in a profit of $4,000.

Step 3: Determine the Break-even Point

The break-even point is when cost equals revenue, so set \( C(q) = R(q) \): \( 6000 + 10q = 12q \). Simplifying gives \( 6000 = 2q \) or \( q = 3000 \). Therefore, the break-even quantity is 3000 units.

Step 4: Graphical Illustration of the Break-even Point

On a graph with quantity on the x-axis and dollars on the y-axis, plot both cost \( C(q) = 6000 + 10q \) and revenue \( R(q) = 12q \). The point where they intersect at \( q = 3000 \) and \( y = 36000 \) represents the break-even point. Below 3000 units, costs are higher than revenue, and above 3000 units, revenue surpasses costs.

Key Concepts

Understanding Cost Function

A cost function is a mathematical expression representing the total cost of production as a function of the quantity of goods produced. The given cost function in our exercise is \( C(q) = 6000 + 10q \). This equation breaks down the total cost into two components:

• The fixed cost: This is the cost incurred regardless of the amount produced, represented here by 6000 dollars.
• The variable cost: This depends on the number of units produced, represented by \(10q\), where \(10\) is the cost per unit.

For example, if we substitute \( q = 500 \), we find \( C(500) = 11000 \) dollars, indicating that producing 500 units costs 11000 dollars. Similarly, for \( q = 5000 \), \( C(5000) = 56000 \) dollars.

Exploring Revenue Function

The revenue function calculates the income generated from selling a certain number of units. In this scenario, the revenue function is \( R(q) = 12q \). This linear equation shows that revenue is directly proportional to the quantity sold:

• \(12\) is the price per unit, indicating revenue grows by 12 dollars for each additional unit sold.

When substituting \( q = 500 \), we obtain \( R(500) = 6000 \) dollars, which means selling 500 units brings in 6000 dollars. Similarly, for \( q = 5000 \), revenue becomes \( R(5000) = 60000 \) dollars.

Profit and Loss Analysis

Profit and loss analysis involves comparing total cost and total revenue to determine if a business is making a profit or incurring a loss. We calculate profit using the formula: \[\text{Profit} = \text{Revenue} - \text{Cost}\]At 500 units:

• Cost: 11000 dollars
• Revenue: 6000 dollars
• Loss: \( 6000 - 11000 = -5000 \) dollars

At 5000 units:

• Cost: 56000 dollars
• Revenue: 60000 dollars
• Profit: \( 60000 - 56000 = 4000 \) dollars

Hence, the company incurs a loss at 500 units but gains profit at 5000 units.

Graphical Illustration of Break-Even Point

Graphing cost and revenue functions helps to visualize the break-even point, which is the quantity where revenues match costs. Let's break down what happens in the graph:

• The x-axis represents the quantity of units produced or sold.
• The y-axis represents money, including both costs and revenues.
• The cost function \(C(q) = 6000 + 10q\) intersects the revenue function \(R(q) = 12q\) at a specific point.
• This intersection is the break-even point at \( q = 3000 \) units, corresponding to \( y = 36000 \) dollars.
• For quantities below 3000, costs are higher than revenues, implying a loss.
• For any quantity above 3000, revenues exceed costs, indicating profit.

Understanding this graphical representation makes it easy to determine the profitability and sustainability of production levels.