Problem 127 - Algebra and Trigonometry

Question

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is 3000 dollar and it costs 3.00 dollar to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

Step-by-Step Solution

Answer

The company must produce and sell more than 1200 packages of stationery each week to generate a profit.

Step 1: Define the variables

Let \(x\) be the number of packages of stationery that need to be produced and sold each week. The revenue made by the company from selling \(x\) packages is \(5.5x\) dollars. The cost of producing \(x\) packages is \(3x\) dollars. The company also has a fixed cost of $3000.

Step 2: Formulate the inequality

For the company to generate profit, revenue must be greater than the total cost. The total cost includes both the fixed cost and the variable cost of producing the packages. The inequality representing the conditions of the problem is \(5.5x > 3x + 3000\).

Step 3: Solve for \(x\)

Subtract \(3x\) from both sides of the inequality to obtain \(2.5x > 3000\). Divide both sides by 2.5 to solve for \(x\). The result is \(x > 1200\).

Step 4: Interpret the result

Since \(x\) represents the number of stationery packages that should be sold for the company to make a profit, it cannot be a fraction. Therefore, \(x\) must be more than 1200. This means the company must produce and sell more than 1200 packages of stationery each week to generate profit.

Key Concepts

Word Problem Strategies

When dealing with word problems in mathematics, it is important to break down the problem into smaller manageable parts. This helps in understanding the problem more clearly and finding the solution efficiently. Here are some strategies to tackle word problems successfully:

• Read the problem carefully and more than once to understand what's being asked.
• Identify key information and underline keywords or numbers.
• Translate the verbal information into mathematical expressions or inequalities.
• Define variables to represent unknown values in the problem.
• Check each step logically to ensure you’re on the right track.

For the stationery problem, we first identified the variables needed to model the linear inequality. We then formulated the conditions that must be met, using these strategies to cover all aspects of the word problem.

Cost and Revenue Analysis

Understanding cost and revenue is essential in any business scenario, such as the stationery example. It involves assessing both fixed and variable costs compared to the revenue to determine profitability. Here is how you can analyze them:

• The **fixed cost** is a constant expense, which, in this case, is \(3000 weekly. It doesn't change with the number of units produced.
• The **variable cost** changes with production levels, i.e., \)3 per stationery package.
• The **selling price** determines how much revenue is generated per package, which is $5.50 here.
• The **total revenue** can be calculated as the selling price multiplied by the number of packages sold, shown as \(5.5x\).
• The **total cost** includes both fixed and variable expenses, which sum up to \(3x + 3000\).

By setting up these expressions, students can formulate inequalities to determine how many products must be sold to cover all costs and make a profit.

Profit Calculation

Profit calculation using linear inequalities involves ensuring that revenue exceeds total costs. Let's break down the steps used in the stationery problem for clarity:

• Begin by setting the inequality where revenue \(5.5x\) must be greater than costs \(3x + 3000\).
• Simplify the inequality: subtract \(3x\) from both sides to isolate terms containing \(x\): \(2.5x > 3000\).
• Solve for \(x\) by dividing both sides by 2.5, resulting in \(x > 1200\).

This means over 1200 units must be sold to generate a profit. Remember, since fractions of packages can't be sold, \(x\) must be an integer. Hence, 1201 packages or more are needed. Writing out these steps helps students understand each phase of profit calculation and reinforces the importance of translating word problems into equations before solving them.