Problem 34 - A Graphical Approach to College Algebra

Question

Exercises \(33-36\) (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 \(\quad\) 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 technician breaks even at 87,500 copies.

Step 1: Expressing the Cost Function

The cost to run the copying service includes the fixed cost of the copier, which is \(3500, and the variable cost for each copy. Since each sheet costs \)0.01, the variable cost for making \(x\) copies is \(0.01x\). Thus, the cost function is calculated as follows: \[ C(x) = 3500 + 0.01x \] where \(C(x)\) is the total cost and \(x\) is the number of copies made.

Step 2: Expressing the Revenue Function

Revenue is generated from each copy made. Since the technician gets paid $0.05 per copy, the revenue for making \(x\) copies is \(0.05x\). Therefore, the revenue function is: \[ R(x) = 0.05x \] where \(R(x)\) is the total revenue and \(x\) is the number of copies made.

Step 3: Finding the Break-even Point

The break-even point occurs where revenue equals cost, i.e., \(R(x) = C(x)\). Set the equations equal to each other: \[ 0.05x = 3500 + 0.01x \] Solve for \(x\): \[ 0.05x - 0.01x = 3500 \] \[ 0.04x = 3500 \] \[ x = \frac{3500}{0.04} \] \[ x = 87500 \] This means the technician must make 87,500 copies for the revenue to equal the cost.

Step 4: Graphing and Interpretation

To graph \(y_1 = C(x)\) and \(y_2 = R(x)\), set the horizontal axis as copies \(x\) and the vertical axis as dollars. Plot the cost function \(C(x) = 3500 + 0.01x\) and the revenue function \(R(x) = 0.05x\) on the same set of axes. The break-even point occurs where the two lines intersect, which is at \((87500, 4375)\). This intersection shows the point where the revenue matches the cost, indicating no loss or profit at 87,500 copies.

Key Concepts

Break-even Point

The break-even point is a crucial concept in cost and revenue analysis. It's the number of units where total revenue equals total costs, making profit neither a gain nor a loss. For our copying service scenario, finding the break-even involves using the cost and revenue functions. Here, the cost function is given by \( C(x) = 3500 + 0.01x \) and the revenue function by \( R(x) = 0.05x \).
To determine the break-even point, set these functions equal:

• Begin with the equation \( 0.05x = 3500 + 0.01x \).
• Rearrange to find \( x \), resulting in \( 0.04x = 3500 \).
• Solve for \( x \) to get \( x = 87500 \).

At 87,500 copies, the costs equal the revenue, and thus, the technician experiences neither profit nor loss.

Cost Function

The cost function represents the total expenses related to producing a certain number of products or services. It comprises both fixed and variable costs.
For the copying service:

• The fixed cost is the initial investment in the copier and service contract, which totals to \\(3500.
• The variable cost is the expense per copy, here \\)0.01 per sheet of paper.

Thus, the cost function \( C(x) = 3500 + 0.01x \) calculates the total cost \((C(x))\) based on the number of copies \((x)\) made. This function helps understand how costs change with production levels, ensuring clear and straightforward tracking of expenses.

Revenue Function

The revenue function depicts the earnings from selling a certain quantity of goods or services. In the context of the copier service:

• Each sheet earns \$0.05.

Therefore, the revenue function is expressed as \( R(x) = 0.05x \), where \( R(x) \) is the total revenue and \( x \) represents the number of copies. Understanding this function is vital as it provides insight into how revenue grows with an increase in sales. It enables the business to plan strategically for profit generation.

Graphical Interpretation

Graphing the cost and revenue functions allows for a visual understanding of their relationship. For the copying service, plot the formulas on the same graph to reveal their intersection — the break-even point.

• The horizontal axis represents the number of copies \((x)\).
• The vertical axis shows the financial values in dollars.

Draw the cost function \( y_1 = C(x) \), a line beginning at \$3500 due to fixed costs and gradually rising with the cost per sheet. Similarly, represent the revenue function \( y_2 = R(x) \), starting at zero and increasing with each copy's revenue.
Where these lines cross, at \((87500, 4375)\), marks the break-even point. This means selling 87,500 copies results in equal costs and revenue, highlighting the significance of achieving this balance for profitability.