Problem 31 - Algebra and Trigonometry

Question

Printing Costs The cost \(C\) of printing a magazine is jointly proportional to the number of pages \(p\) in the magazine and the number of magazines printed \(m .\) (a) Write an equation that expresses this joint variation. (b) Find the constant of proportionality if the printing cost is \(\$ 60,000\) for 4000 copies of a 120 -page magazine. (c) How much would the printing cost be for 5000 copies of a 92 -page magazine?

Step-by-Step Solution

Answer

(a) \( C = k \cdot p \cdot m \); (b) \( k = \frac{1}{8} \); (c) \$57,500.

Step 1: Understanding Joint Variation

Joint variation implies that a variable varies with the product of two or more variables. In this case, the cost \( C \) is jointly proportional to the number of pages \( p \) and the number of magazines printed \( m \). The equation is expressed as \( C = k \cdot p \cdot m \), where \( k \) is the constant of proportionality.

Step 2: Finding the Constant of Proportionality

To find the constant \( k \), use the given information: the printing cost \( C \) is \$60,000 for 4000 copies of a 120-page magazine. Set up the equation: \( 60000 = k \cdot 120 \cdot 4000 \). Simplify to find \( k \):\[ k = \frac{60000}{120 \times 4000} \]Calculate \( k \):\[ k = \frac{60000}{480000} = \frac{1}{8} \]

Step 3: Calculating the Printing Cost for Different Quantities

Now, substitute the constant \( k = \frac{1}{8} \) into the equation \( C = k \cdot p \cdot m \) to find the cost for 5000 copies of a 92-page magazine. Substitute the values: \( p = 92 \) and \( m = 5000 \):\[ C = \frac{1}{8} \times 92 \times 5000 \]Calculate the product:\[ C = \frac{1}{8} \times 460000 = 57500 \]The printing cost would be \$57,500.

Key Concepts

Understanding the Constant of Proportionality

In mathematical relationships, the constant of proportionality serves as a crucial factor. It represents the unchanging numerical value in a proportional relationship. In the context of our exercise, it connects the printing cost to the measurable factors: number of pages and number of magazines.

This constant helps us understand how much one unit of the product affects the outcome. When determining the constant of proportionality, we'd first express the joint variation equation as:

• \( C = k \cdot p \cdot m \)

Here, \( C \) is the cost, \( p \) represents the number of pages, \( m \) is the number of magazines, and \( k \) is the constant of proportionality.

By solving for \( k \), we can analyze the relationship further and perform cost calculations with various values of \( p \) and \( m \). In our example, \( k \) was calculated as \( \frac{1}{8} \). This means each unit product of pages and magazines contributes this constant amount to the total cost.

Understanding \( k \) ensures our proportional relationships are exact and flexible to meet different needs.

Exploring Proportional Relationships

A proportional relationship shows how two quantities vary together, maintaining a constant ratio. It's like a balanced seesaw: when one side rises, the other follows proportionally. This concept plays a pivotal role in our exercise, focusing on joint variation.

Joint variation means the cost, \( C \), is influenced by two factors simultaneously: number of pages, \( p \), and number of magazines, \( m \). We express this relation as:

• \( C = k \cdot p \cdot m \)

This equation embodies the direct proportionality of the cost to both variables, \( p \) and \( m \), interacted through \( k \).

If either \( p \) or \( m \) increases, holding the other constant, the cost will also increase in a predictable manner. By understanding this predictable change, it becomes easier to project costs under varying production conditions. This knowledge is practical for businesses, allowing them to estimate expenses efficiently and effectively.

Performing Cost Calculations

Cost calculations using joint variation provide a structured way to determine expenses in business operations. By manipulating the variables we already know, we can anticipate costs and budget accordingly.

In our exercise, the cost, \( C \), is determined by the equation:

• \( C = k \cdot p \cdot m \)

Once we find the constant \( k \), the calculation becomes straightforward. Given \( k = \frac{1}{8} \) from our example, we can calculate costs for any combination of \( p \) and \( m \).

For instance, the cost of producing 5000 copies with 92 pages each was calculated as follows:

• \( C = \frac{1}{8} \times 92 \times 5000 = 57500 \)

Therefore, the expense for this production scenario would be $57,500.

Understanding how to perform these calculations allows businesses to flexibly allocate resources and predict financial outcomes, ensuring smoother operational planning.