Problem 34 - 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. Copier Service A technician runs a copying service in his home. He paid \(\$ 3500\) for the copier and a lifetime service contract. Each sheet of paper costs \(\$ 0.01,\) and he gets paid \(\$ 0.05\) per copy. Let \(x\) be the number of copies he makes.

Step-by-Step Solution

Answer

The breakeven point is 87,500 copies.

Step 1: Define the Cost Function

The total cost consists of the initial fixed cost and the variable cost per copy. The fixed cost is \(3500, and the variable cost is \)0.01 per copy. Therefore, the cost function can be expressed as:\[ C(x) = 3500 + 0.01x \]

Step 2: Define the Revenue Function

Revenue is generated by charging $0.05 per copy. Thus, the revenue function is defined as:\[ R(x) = 0.05x \]

Step 3: Find the Breakeven Point

The breakeven point occurs when cost equals revenue, i.e., \( C(x) = R(x) \). Set the cost function equal to the revenue function and solve for \( x \):\[ 3500 + 0.01x = 0.05x \]Subtract \(0.01x\) from both sides:\[ 3500 = 0.04x \]Divide both sides by 0.04 to find \(x\):\[ x = \frac{3500}{0.04} = 87500 \]

Step 4: Graph the Functions and Interpret

Plot \(y_1 = C(x) = 3500 + 0.01x\) and \(y_2 = R(x) = 0.05x\) on the same set of axes. The line representing \(C(x)\) starts at $3500 on the y-axis, since this is the fixed cost, and has a shallower slope compared to \(R(x)\). The lines intersect at \(x = 87500\), showing that this is the point where revenue equals cost. Beyond this point, revenue exceeds cost, indicating profit.

Key Concepts

Cost Function

A cost function is a mathematical formula used to calculate the total expense incurred in the production of goods or services. In any business, especially a service or manufacturing setup, understanding costs is crucial for decision-making and financial planning.

For the cost function, we consider both fixed and variable costs:

• Fixed Costs: These are expenditures that do not change with the level of production, such as rent, salaries, and equipment costs. In the exercise, the fixed cost of \(3500 accounts for the purchase and maintenance of the copier.
• Variable Costs: These costs vary directly with the production volume. In our example, the variable cost is \)0.01 per copy.

Combining both gives us the cost function: \[ C(x) = 3500 + 0.01x \]This formula enables the calculation of total costs for any number of copies, making planning and budgeting more effective. For instance, if he makes 1,000 copies, the cost would be:\[ C(1000) = 3500 + 0.01 \times 1000 = 3510 \]

Revenue Function

Revenue is the income generated from selling goods or services. The revenue function helps businesses track earnings as they produce and sell more units. It is essential to distinguish the revenue generated per unit from the overall process.

In the given problem, each copy earns the technician \(0.05. Thus, the revenue function can be expressed as:\[ R(x) = 0.05x \]Where:

• x is the number of copies made, and
• \)0.05 is the revenue per copy.

For example, if the technician makes 1,000 copies, the revenue would be:\[ R(1000) = 0.05 \times 1000 = 50 \]Having a clear understanding of the revenue function allows businesses to forecast future earnings and adjust strategies as required.

Breakeven Point

Understanding the breakeven point is central to any financial analysis for a business. The breakeven point is where total revenues and total costs are equal, meaning no profit or loss is being made. It's a critical milestone that indicates how much production is necessary to start earning a profit.

To find this point, set the cost function equal to the revenue function:\[ C(x) = R(x) \]Using the provided functions:\[ 3500 + 0.01x = 0.05x \]Solving for x, we rearrange and simplify:\[ 3500 = 0.04x \]\[ x = \frac{3500}{0.04} = 87500 \]At this point, making 87,500 copies, the technician covers all costs including fixed and variable. Producing more than this quantity will result in profit, making it a valuable turning point.

Graphing Functions

Graphing functions visually represents relationships between variables, offering clear insights that are sometimes missed in pure numerical form. In this exercise, graphing both cost and revenue functions on the same axes can help visualize when revenue surpasses costs.

For the functions:

• C(x): Starts at a point representing the fixed cost ($3500) on the y-axis and ascends gradually with a slope of 0.01, reflecting the variable cost per copy.
• R(x): Begins at the origin as it solely depends on sales volume and climbs faster, having a steeper slope of 0.05.

The intersection point of the two lines on the graph is significant. It occurs at (87,500, 4375), pinpointing the breakeven point. Beyond this point, the revenue line lies above the cost line, indicating profitable territory.

This graphical representation helps stakeholders make informed decisions about scaling production to ensure profitability.