Problem 33 - Intermediate Algebra

Question

Publishing. \(\quad\) A printer has two presses. The older press has a setup cost of \(\$ 210\) and can print the pages of a certain book for S5.98. The newer press has a setup cost of \(\$ 350\) and can print the pages of the same book for \(\$ 5.95\) a. Find the break point. b. If the publisher has advance orders for \(5,100\) copies of the book, which press should be used?

Step-by-Step Solution

Answer

The break point is 4,667 copies; use the newer press for 5,100 copies.

Step 1: Define the Cost Equations for Each Press

Let \( x \) be the number of books printed. For the older press, the total cost \( C_1 \) is given by \[C_1 = 210 + 5.98x\]For the newer press, the total cost \( C_2 \) is given by \[C_2 = 350 + 5.95x\]

Step 2: Find the Break Point

The break point occurs where the cost of using each press is equal. Set \( C_1 = C_2 \):\[210 + 5.98x = 350 + 5.95x\]Solving for \( x \) gives:\[210 - 350 = 5.95x - 5.98x\]\[-140 = -0.03x\]\[x = \frac{140}{0.03} = 4666.67\]Since the number of books must be whole, the break point is at \( x = 4667 \).

Step 3: Determine Which Press to Use for 5,100 Copies

Calculate the total cost for each press at \( x = 5100 \):For the older press:\[C_1 = 210 + 5.98(5100) = 30,708\]For the newer press:\[C_2 = 350 + 5.95(5100) = 30,695\]Since \( C_2 < C_1 \), the newer press should be used for 5,100 copies.

Key Concepts

Cost Functions

Understanding cost functions is crucial in assessing the financial aspects of business operations, especially in decision-making scenarios like this printing problem. A cost function is a mathematical equation that describes how the total cost of production varies with the level of output. In this exercise, we define separate cost functions for each press.

The older press has a setup cost of \( \\(210 \) plus a variable cost of \( \\)5.98 \) per book. The total cost function for it becomes:

• \( C_1 = 210 + 5.98x \)

Similarly, the newer press has a setup cost of \( \\(350 \), with a slightly lower variable cost of \( \\)5.95 \) per book. Thus, its total cost function is:

• \( C_2 = 350 + 5.95x \)

The setup cost represents a fixed cost paid regardless of the number of books printed, while the variable cost depends directly on the quantity \( x \). These cost functions allow us to calculate the total cost for any number of books.

Equation Solving

Once the cost functions are set up, we need to solve equations to find the break-even point where the costs of using each press are equal. This break-even analysis helps in determining the point at which it becomes more cost-effective to switch from the older press to the newer one.

Let's set up the equation by equating the two cost functions:

• \( 210 + 5.98x = 350 + 5.95x \)

Now, we solve for \( x \) (the number of books):

• Subtracting \( 210 \) and \( 350 \), we get \( 210 - 350 = 5.95x - 5.98x \)
• This simplifies to \(-140 = -0.03x \)
• Dividing both sides by \(-0.03\), we find \( x = \frac{140}{0.03} \)

This computation leads to \( x = 4666.67 \). Since we can't print a fraction of a book, we round to the nearest whole number, resulting in a break-even point of \( x = 4667 \) books.

Comparative Cost Analysis

With the break-even point known, we can use comparative cost analysis to evaluate which press to use when specific quantities of books are ordered. This analysis helps us choose the option that minimizes cost, thereby maximizing profit.

For 5,100 copies, we calculate the total cost for each press:

• Older Press: \( C_1 = 210 + 5.98(5100) = 30,708 \)
• Newer Press: \( C_2 = 350 + 5.95(5100) = 30,695 \)

Comparing these results, the newer press offers a lower total cost at \( \\(30,695 \) compared to \( \\)30,708 \) for the older press.

This cost-saving difference grows with larger orders because of the smaller variable cost of the newer press. Conducting such a comparative cost analysis allows businesses to make informed decisions based on cost efficiencies, which is crucial for financial optimization.