Problem 25 - 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

The cost for 5000 copies of a 92-page magazine is \(\$57,500\).

Step 1: Understanding Joint Variation

When a quantity varies jointly with two other quantities, it is proportional to the product of those quantities. This means that the cost of printing \( C \) is proportional to the product of the number of pages \( p \) and the number of magazines \( m \). The equation expressing this relationship is:\[ C = k \cdot p \cdot m \]where \( k \) is the constant of proportionality.

Step 2: Finding the Constant of Proportionality

Given that the printing cost is \( \$60,000 \) for 4000 copies of a 120-page magazine, substitute these values into the equation to find \( k \):\[ 60000 = k \cdot 120 \cdot 4000 \]\[ 60000 = k \cdot 480000 \]Divide both sides by 480000 to solve for \( k \):\[ k = \frac{60000}{480000} = 0.125 \]

Step 3: Calculating the Cost for Different Conditions

Now that we know \( k = 0.125 \), use this value to find the cost of printing 5000 copies of a 92-page magazine. Substitute \( p = 92 \), \( m = 5000 \), and \( k = 0.125 \) into the equation:\[ C = 0.125 \cdot 92 \cdot 5000 \]Calculate the product:\[ C = 0.125 \cdot 460000 = 57500 \]

Key Concepts

constant of proportionality

The constant of proportionality, often denoted as \( k \), is a crucial concept in understanding joint variation. In a relationship where one quantity is jointly proportional to two or more other quantities, \( k \) acts as a multiplier that adjusts the scale of this proportional relationship. It determines how much the output varies relative to the input quantities.
In the given exercise, the printing cost \( C \) varies jointly with the number of pages \( p \) and the number of magazines printed \( m \). Thus, \( k \) will define how these inputs are scaled to determine the cost. To find \( k \), we need specific instances where all variable values are known. For example, if 4000 magazines of 120 pages each cost \(60,000 to print, substituting these values into the joint variation equation \( C = k \cdot p \cdot m \) allows us to solve for \( k \).
The formula becomes \( 60000 = k \cdot 120 \cdot 4000 \). Solving, \( k \) equals \( 0.125 \), which means for these particular conditions, every set unit of pages and copies costs \)0.125 to print.

proportional relationships

Proportional relationships are connections between quantities where one quantity is a constant multiple of the other. When a relationship is proportional, any change in one quantity results in a directly related change in the other quantity.

• Direct Proportionality: If something increases, the other also increases and vice versa.
• Joint Proportionality: Involves more than one quantity influencing the proportional relationship.

In the exercise scenario, the proportionality is described as joint because the printing cost \( C \) depends directly on both the number of pages \( p \) and the number of magazines \( m \). This creates a three-variable relationship where changing either \( p \) or \( m \) directly influences \( C \), adjusted by the constant \( k \).
The key here is understanding that both independent quantities (pages and magazines) multiply together to affect the cost. This compounded effect is what makes joint proportionality a powerful tool for scaling and predicting costs in scenarios like printing.

equation for joint variation

The equation for joint variation directly defines how one variable depends on the product of two or more other variables. In our exercise, the equation \( C = k \cdot p \cdot m \) models the relationship between the cost \( C \), the number of pages \( p \), and the number of magazines \( m \). This simple equation captures the essence of joint variation.

• The variable \( C \) represents what you're trying to find; in this case, the printing costs.
• \( p \) and \( m \) are your influencing variables, pages and magazines, respectively.
• \( k \) is the constant reflecting the proportional "weight" or cost per unit increase of \( p \) and \( m \).

To effectively use this equation, remember these steps:1. Substitute the known values into the equation to find \( k \) when given all other variables.2. Use this \( k \) to predict outcomes for different scenarios where \( p \) or \( m \) change. For instance, by knowing \( k = 0.125 \), the cost equation becomes precise, and we can easily determine that printing 5000 copies of a 92-page magazine will cost $57,500 by substituting these new values into the equation.