Problem 15 - Applied Calculus

Question

A company that makes Adirondack chairs has fixed costs of $$ 5000$ and variable costs of $$ 30 per chair. The company sells the chairs for $$ 50 each. (a) Find formulas for the cost and revenue functions. (b) Find the marginal cost and marginal revenue. (c) Graph the cost and the revenue functions on the same axes. (d) Find the break-even point.

Step-by-Step Solution

Answer

The break-even point is at 250 chairs.

Step 1: Establish the Cost Function

The total cost function consists of fixed costs and variable costs. The fixed cost is given as \(5000, and the variable cost per chair is \)30. Therefore, the cost function is given by:\[ C(x) = 5000 + 30x \]where \(x\) represents the number of chairs produced.

Step 2: Establish the Revenue Function

The revenue function is the product of the number of chairs sold and the price per chair. The price per chair is $50, so the revenue function is:\[ R(x) = 50x \]where \(x\) is the number of chairs sold.

Step 3: Calculate Marginal Cost

The marginal cost is the derivative of the cost function with respect to \(x\). Since the cost function is \(C(x) = 5000 + 30x\), its derivative is:\[ C'(x) = 30 \]Thus, the marginal cost is $30 per chair.

Step 4: Calculate Marginal Revenue

The marginal revenue is the derivative of the revenue function with respect to \(x\). Given \(R(x) = 50x\), its derivative is:\[ R'(x) = 50 \]Thus, the marginal revenue is $50 per chair.

Step 5: Graph the Cost and Revenue Functions

To graph \(C(x) = 5000 + 30x\) and \(R(x) = 50x\), plot these linear functions on the same set of axes with \(x\) (number of chairs) on the horizontal axis and dollars () on the vertical axis. The \(C(x)\) starts at 5000 when \(x=0\), while \(R(x)\) starts at the origin.

Step 6: Find the Break-even Point

The break-even point occurs where cost equals revenue, i.e., \(C(x) = R(x)\). Set the equations equal to find \(x\):\[ 5000 + 30x = 50x \]Simplify to obtain:\[ 5000 = 20x \]\[ x = 250 \]Thus, the break-even point is at 250 chairs, where revenue equals cost.

Key Concepts

Revenue Functions

Revenue functions are an important concept in business, representing the total income a company earns from selling its products. It is fundamentally based on the product of the quantity sold and the price per product. In the context of our Adirondack chairs example, if the company sells each chair for \(50\). This means the revenue function would be:

• \(R(x) = 50x\)

Where \(x\) indicates the number of Adirondack chairs sold.
The revenue function helps businesses predict total sales for varying levels of production and can guide pricing and production decisions.

Marginal Cost

Marginal cost is a key concept in economic theory and operations management, representing the increase in total cost that arises from producing one additional unit of a product. For businesses, understanding marginal cost is crucial for optimizing production levels and setting optimal pricing. In our example, the cost function is defined as:

• \(C(x) = 5000 + 30x\)

This function incorporates both the fixed costs (\(5000\)) and variable costs (\(30\) per chair). The marginal cost is the derivative of the total cost function, which in this case, simplifies to:

• \(C'(x) = 30\)

This means that each additional chair incurs a cost of \(\$30\), helping the company decide how many chairs to produce to manage profitability effectively.

Break-even Point

The break-even point is critical for any business, indicating where total costs equal total revenues, resulting in neither profit nor loss. It's the threshold after which a company begins to see actual profits from its sales. For our Adirondack chairs, the break-even point can be found by setting the revenue and cost functions equal:

• \(5000 + 30x = 50x\)

After solving this equation, we find:

• \(x = 250\)

This means the company needs to sell 250 chairs to cover all its costs. Beyond this point, every chair sold contributes to the profit. Knowing the break-even point helps businesses make informed operational decisions and understand the level of sales needed to sustain the business.

Graphing Linear Functions

Graphing linear functions assists in visualizing relationships between variables, which is beneficial for interpreting cost and revenue scenarios in a business. Both cost and revenue functions can be graphed as straight lines. The cost function \(C(x) = 5000 + 30x\) will start at \(5000\) on the y-axis, reflecting the fixed costs when no chairs are sold, and will increase at a rate of \(30\) per chair. It shows each chair increases costs linearly.
In contrast, the revenue function \(R(x) = 50x\) begins at the origin, indicating no revenue at zero sales. As the company sells more chairs, this line will slope upwards, steepening more quickly than the cost line due to the \(50\) revenue per chair.
By plotting these on the same graph, intersection points like the break-even point can clearly be observed, where cost and revenue lines cross, elucidating where the company starts making a profit. Graphs offer an intuitive and immediate understanding of how products and pricing interact with cost factors.