Problem 50 - Precalculus : Building Concepts and Connections

Question

To print booklets, it costs 400 dollars plus an additional 0.50 dollars per booklet. What is the minimum number of booklets that must be printed so that the average cost per booklet is less than 0.55 dollars?

Step-by-Step Solution

Answer

The minimum number of booklets that must be printed so that the average cost per booklet is less than 0.55 dollars is 4001.

Step 1: Define the Average Cost Equation

First, the average cost of the booklet can be found by dividing the total cost of production by the number of booklets. Formulate the average cost (AC) equation as, \( AC = \rac{{400 + 0.50n}}{n} \), where n represents the number of booklets.

Step 2: Set Up the Inequality

Since the problem is asking for the minimum number of booklets for which the average cost will be less than $0.55, you need to set an inequality. The inequality to solve is \( \rac{{400 + 0.50n}}{n} < 0.55 \). This inequality means the average cost of producing n booklets has to be less than $0.55.

Step 3: Solve the Inequality

Next, one should solve the inequality for n. To do this, multiply each side by n to clear the denominator and then solve the ensuing equation. Doing this, the inequality becomes \( 400 + 0.50n < 0.55n \). Solving this gives \( n > 4000 \).

Step 4: Round Up the Result

Since the number of booklets cannot be a fraction, the number of booklets must be rounded up to the nearest integer. In this case, even though \( n > 4000 \), since we cannot produce fractional booklets, the minimum number that meets the condition will be 4001.

Key Concepts

Average Cost

The average cost is an important concept in understanding economics and making efficient decisions in production. Average cost refers to the cost per unit of output. It is calculated by dividing the total cost by the number of units produced or sold. In the context of the problem, we're looking at the average cost per booklet. We begin by computing the total cost, including both fixed and variable costs.

• Fixed Costs: In our problem, fixed cost is \(400, which must be paid regardless of the number of booklets printed.
• Variable Costs: Each booklet has an additional cost of \)0.50, making it a variable cost that changes with the number of booklets.

To compute the average cost (AC), we use the formula: \[ AC = \frac{{400 + 0.50n}}{n} \] where \( n \) is the number of booklets. This formula shows how the cost behavior changes with variations in \( n \). As production increases, fixed costs are spread across more booklets, potentially lowering the average cost.

Production Cost

Production cost is the combination of all costs a company incurs to produce a particular amount of products. Understanding these costs helps businesses in budget planning and price setting. In our problem, we see two components that make up the production cost:

• Fixed Production Cost: This is a constant cost incurred by the printing company, which is \(400 for the booklets. It doesn't change with production volume.
• Variable Production Cost: This cost increases with the number of booklets produced at \)0.50 per booklet. Thus, if you produce more, this cost rises proportionally.

To determine the total production cost, you need to sum both costs:
\[ ext{Total Cost} = 400 + 0.50n \] For smart production management, it's crucial to understand how these costs influence pricing and profitability, aiding in financial strategizing and decision-making.

Problem Solving

Problem solving in mathematics often involves logical thinking and methodical steps to reach a solution. In tackling inequalities like the one in the problem, we take systematic steps:

• First, we set up an equation or inequality using the given data. For this problem, it was ensuring the average cost is less than \(0.55.
• Then, we manipulate the inequality to isolate the variable of interest, \( n \) in this case.
• The inequality is modified from \( \frac{{400 + 0.50n}}{n} < 0.55 \) to \( 400 + 0.50n < 0.55n \).
• Next, solve for \( n \), giving \( n > 4000 \).
• Finally, because items like booklets must be whole numbers, we round up to ensure the constraint (average cost under \)0.55) is met, resulting in \( n = 4001 \).

Effective problem solving requires breaking the problem down, evaluating each step carefully, and ensuring practical solutions that can be implemented, such as considering whole units for production.