Problem 35 - A Graphical Approach to Precalculus with Limits

Question

do each of the following. (a) Express the cost \(C\) as a function of \(x,\) where \(x\) represents the number of items as described. (b) Express the revenue \(R\) as a function of \(x .\) (c) Determine analytically the value of \(x\) for which revenue equals cost. (d) Graph \(y_{1}=C(x)\) and \(y_{2}=R(x)\) on the same \(x y\) -axes and interpret the graphs. Delivery Service \(A\) truck driver operates a delivery ervice. His start-up costs amounted to \(\$ 2300 .\) He estimates that it costs him (in terms of gasoline, wear and tear =on his truck, etc.) \(\$ 3.00\) per delivery. He charges \(\$ 5.50\) per delivery. Let \(x\) represent the number of deliveries he makes.

Step-by-Step Solution

Answer

The break-even point occurs at 920 deliveries.

Step 1: Expressing Cost Function C(x)

The total cost \( C(x) \) consists of a start-up cost and a variable cost per delivery. The start-up cost is \( 2300 \) dollars, and the variable cost is \( 3.00 \) dollars per delivery. Thus, the cost function is given by the expression:\[ C(x) = 2300 + 3x \]where \( x \) is the number of deliveries.

Step 2: Expressing Revenue Function R(x)

The revenue function \( R(x) \) is calculated by multiplying the charge per delivery by the number of deliveries. Since the charge is \( 5.50 \) dollars per delivery, the revenue function is defined as:\[ R(x) = 5.5x \]where \( x \) is the number of deliveries.

Step 3: Finding Break-even Point

To find the break-even point, we set the cost function \( C(x) \) equal to the revenue function \( R(x) \). Equating them gives:\[ 2300 + 3x = 5.5x \] Simplifying the equation, we find:\[ 2300 = 2.5x \] Solving for \( x \) gives:\[ x = \frac{2300}{2.5} = 920 \] deliveries.

Step 4: Graphing Functions and Interpretation

Plot the graphs of \( y_1 = C(x) = 2300 + 3x \) and \( y_2 = R(x) = 5.5x \) on the same set of axes. The y-axis represents dollars, and the x-axis represents the number of deliveries. The graph of \( C(x) \) is a straight line starting at 2300 on the y-axis with a slope of 3, while \( R(x) \) is a straight line starting at the origin with a slope of 5.5. The intersection of these lines at \( x = 920 \) indicates the break-even point, where revenue equals cost.

Key Concepts

Cost Function

The cost function is a critical tool in break-even analysis, as it helps identify the total cost of operating a business up to a certain point. In this exercise, the cost function is used to calculate the total expenses incurred by a truck driver providing delivery services. The truck driver's start-up cost is a fixed cost of \(2300, representing initial investments such as securing a vehicle. Additionally, there are variable costs associated with each delivery, which in this case, is \)3.00 per delivery. Together, these costs form the cost function:\[ C(x) = 2300 + 3x \]Where:

• \(2300\) is the fixed startup cost in dollars.
• \(3\) is the cost per delivery in dollars.
• \(x\) represents the number of deliveries made.

Understanding the cost function allows the driver to predict how expenses will rise with an increase in deliveries, offering vision into managing and forecasting future costs.

Revenue Function

Revenue function is another crucial element in financial analysis, representing the total income earned through business operations. For the delivery service in this scenario, revenue is generated by charging per delivery.The driver earns $5.50 for each delivery completed. Thus, the revenue function is expressed as:\[ R(x) = 5.5x \]Where:

• \(5.5\) is the earnings from each delivery in dollars.
• \(x\) is the number of deliveries made.

This function helps ascertain how much money the driver will make as the number of deliveries increases. By comparing this with the cost function, the driver can smartly plan for a profitable operation. Understanding revenue functions is key to assessing business efficiency.

Graph Interpretation

Graph interpretation is a vital skill in understanding business operations and improving decision-making processes. The graphical representation of the cost and revenue functions provides a visual understanding of the relationship between deliveries, costs, and revenue.In the graph:

• The line representing the cost function \(y_1 = C(x) = 2300 + 3x\) starts at 2300 (y-intercept), reflecting the fixed startup cost, and has a slope of 3, showing the incremental cost per delivery.
• The line depicting the revenue function \(y_2 = R(x) = 5.5x\) originates from the origin with a slope of 5.5, indicating earnings per delivery.

These lines intersect at \(x = 920\), the break-even point, where revenue matches costs. It means at 920 deliveries, the business neither makes a profit nor incurs a loss. Beyond this point, earnings will surpass expenses, paving the way to profitability. Graphs provide an intuitive comprehension of financial thresholds and guide strategic business adjustments.