Problem 106 - College Algebra

Question

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it cost \(\$ 3.00\) 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 have a profit gain?

Step-by-Step Solution

Answer

The company needs to produce and sell more than 1200 packages of stationery each week in order to see a profit.

Step 1: Understand the Problem

The weekly fixed cost for the company is \$3000. This is a constant value which is always subtracted from the revenue to obtain the net income. The cost to manufacture each stationery package is \$3.00 and they are sold for \$5.50 each. The company sees a profit when the income from selling packages surpasses the total cost (fixed cost + production cost). The number of packages sold per week that needs to be calculated for profit gain.

Step 2: Determine the Revenue and Cost Functions

Let \(x\) represent the number of packages produced and sold each week. Therefore, the revenue from selling \(x\) packages would be \(5.50x\) (selling price multiplied by quantity sold). Similarly, the cost for producing \(x\) packages is \(3x\) (production price multiplied by quantity). The total cost would be the sum of the fixed cost and the variable cost, resulting in \(3000 + 3x\).

Step 3: Setting up the Inequality

The company gains profit when its total revenue is greater than its total cost, which gives an inequality \(5.50x > 3000 + 3x\). This inequality needs to be solved for \(x\) to find out the minimum number of packages to be sold for profit gain.

Step 4: Solve the Inequality

Solving the inequality \(5.50x > 3000 + 3x\) involves first simplifying it by subtracting \(3x\) from both sides giving \(2.50x > 3000\). Then to solve for \(x\), divide both sides by \(2.50\) to get \(x > 1200\).

Key Concepts

Linear Inequality

In the world of word problems, linear inequalities help us express situations where there are constraints or conditions that need to be satisfied. When we say "linear," we're talking about straight-line relationships, meaning no variables are squared or multiplied together. An inequality means that these relationships are not equal–one side is "greater than" or "less than" the other.

In our stationery word problem, the inequality comes into play when deciding the number of packages that must be sold to achieve profitability. The linear inequality that models this situation is:

• Profit inequality: \( 5.50x > 3000 + 3x \)

This says that the revenue must be larger than the cost to have a profit. Understanding how to create and solve a linear inequality is crucial in making decisions like how much product to manufacture or the minimum sales needed to cover costs.

Cost Function

A cost function gives us a mathematical way to look at all the expenses involved in producing a product. For word problems like the stationery example, it helps us calculate the total cost, which includes both fixed and variable costs.

Fixed costs are expenses that don't change, even if production increases or decreases. In this case, the fixed cost is:

• \(\\(3000 \) per week

The variable costs depend on the quantity of products produced. For each package of stationery, this cost was \(\\)3\). The cost function for producing \(x\) packages is:

• \( Cost = 3000 + 3x \)

Combining fixed and variable costs helps determine the break-even point and when profit begins.

Revenue Function

The revenue function tells you how much money you bring in from selling your products. The key is to know your selling price per unit and the quantity sold. In this exercise, the selling price per package is \(\$5.50\). So, the revenue function is about multiplying the number of packages sold by the selling price.

The revenue function for \(x\) packages would be:

• \( Revenue = 5.50x \)

Revenue functions are essential for understanding how sales numbers impact overall finances. The goal is often to maximize revenue by improving sales strategies, all while keeping costs in check.

By comparing the revenue function to the cost function, businesses can assess whether they are making a profit or need to adjust their production or sales strategies.

Profit Calculation

Profit calculation is fundamental in determining a company's financial health. Simply put, profit is what's left after subtracting the costs from the revenue.

In our stationery problem, the equation for profit is:

• Profit = Revenue - Cost

Businesses use this formula to evaluate their financial success. The stationery company gained profit when \(5.50x\) (the revenue) is greater than \(3000 + 3x\) (the total cost).

Solving for \(x\), we:

• Rearrange the inequality: \(2.50x > 3000\)
• Solve for \(x\): \(x > 1200\)

This means selling more than 1200 packages per week leads to a profit. Understanding profit calculations allows businesses to make informed decisions on pricing, production, and sales volume.