Problem 57 - College Algebra

Question

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for \(\$ 86\) each, with a delivery fee of \(\$ 9,200\), regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are sold, how much should they charge for the vacuums?

Step-by-Step Solution

Answer

The store should charge $126 for each vacuum to start seeing a profit after selling 230 units.

Step 1: Define the Variables

Let's define the variables we will use in this problem. Let \( x \) represent the number of vacuums sold, and \( y \) represent the price at which each vacuum is sold.

Step 2: Set Up the Cost Equation

The total cost (\( C \)) of purchasing \( x \) vacuums includes the cost of the vacuums themselves and the delivery fee. Thus, the cost is given by the equation: \[ C = 86x + 9200 \].

Step 3: Set Up the Revenue Equation

The total revenue (\( R \)) from selling \( x \) vacuums at \( y \) dollars each is given by the equation: \[ R = xy \].

Step 4: Determine Break-even Condition

To start being profitable after 230 units are sold, the revenue must equal or exceed the cost for \( x = 230 \) units, or \[ 230y = 86(230) + 9200 \].

Step 5: Solve for Selling Price

Substitute \( x = 230 \) into the break-even equation: \[ 230y = 19780 + 9200 \]. Calculate the total cost: \[ 19780 + 9200 = 28980 \]. Solve for \( y \): \[ y = \frac{28980}{230} = 126 \].

Key Concepts

Augmented Matrix

An augmented matrix is a concept used in linear algebra to represent systems of linear equations. This kind of matrix holds both the coefficients of the variables and the constants, allowing easy manipulation and solution of the system.
For instance, a system of two equations in two variables would get arranged into a form of an augmented matrix that simplifies computation using row operations or software.

• This method can streamline the computation by reducing the need for substitution or elimination methods.
• It is a foundational tool in solving simultaneous equations, especially when automated tools are used for processing.

In our appliance store scenario, setting up an augmented matrix could assist in organizing and solving the equations related to cost and revenue, although the process described doesn't directly employ this technique.
Ultimately, matrices are incredibly powerful in linear algebra, allowing quick adjustments, such as testing different selling prices or costs.

Break-even Analysis

The break-even analysis is a financial assessment that determines when a company will start to generate a profit. It helps identify the point at which revenue will equal costs, allowing a business to start making money beyond expenses.
In practice, the break-even point is calculated by comparing total expenses and earnings.

• This technique is crucial for business planning and decision-making.
• Understanding this concept can help businesses make informed pricing decisions.

In the problem mentioned, the appliance store needs to find out the selling price for vacuums such that they achieve break-even status after selling 230 units.
This involves setting the revenue expression equal to the cost expression and solving for the selling price, ensuring that profits can begin after surpassing this sales volume threshold.

Revenue Equation

The revenue equation calculates the total income or revenue a business generates from selling a number of products at a particular price. The general form is simple and direct: Revenue (\( R \)) equals the unit price (\( y \)) multiplied by the number of units sold (\( x \)).

• The revenue equation helps businesses project future income and make strategic pricing decisions.
• It is essential for assessing profitability and pricing strategy.

In the appliance store's situation, the revenue equation \( R = xy \) represents selling \( x \) vacuums, each priced at \( y \).
It's vital to ensure that this calculated revenue, when equal to the cost, identifies the break-even point where profits may begin accruing.

Cost Equation

The cost equation is used to determine the total expenses associated with a business activity, comprising both variable and fixed costs. The principle formula here involves adding the cost of production per unit by the number of units to any fixed costs the business incurs. In our case, the appliance store's total cost is formulated as \( C = 86x + 9200 \).

• The term 86x represents the variable cost of producing each vacuum at \\(86 per unit.
• The amount \\)9200 stands for fixed costs like delivery fees.

Understanding the cost equation helps set an appropriate price, where sales can occur without a loss.
It aligns crucially with the break-even analysis, enabling businesses to strategize around pricing to cover all expenses comprehensively.