Problem 56 - College Algebra with Corequisite Support

Question

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. Bikes'R'Us manufactures bikes, which sell for \(\$ 250\). It costs the manufacturer \(\$ 180\) per bike, plus a startup fee of \(\$ 3,500\) After how many bikes sold will the manufacturer break even?

Step-by-Step Solution

Answer

The manufacturer breaks even after selling 50 bikes.

Step 1: Understanding the Problem

We need to determine the number of bikes the manufacturer must sell to cover all the costs. The manufacturer incurs a fixed cost of $3500 and a variable cost of $180 per bike, while the selling price per bike is $250.

Step 2: Set Up the Equations

To break even, the total cost must equal the total revenue. Let \( x \) represent the number of bikes sold. The revenue from selling \( x \) bikes is \( 250x \), and the total cost for producing \( x \) bikes is \( 180x + 3500 \). The equation for break even is: \[ 250x = 180x + 3500 \]

Step 3: Form the Augmented Matrix

Rewrite the equation in a form suitable for a matrix. Subtract \( 180x \) from both sides to get: \[ 70x = 3500 \] This can be represented as the augmented matrix: \[ \begin{bmatrix} 70 & | & 3500 \end{bmatrix} \]

Step 4: Solve for x

To solve the matrix equation \( 70x = 3500 \), divide both sides of the equation by 70 to isolate \( x \): \[ x = \frac{3500}{70} \] Carry out the division to find \( x \).

Step 5: Conclusion

Perform the division \( 3500 ÷ 70 = 50 \). Therefore, the manufacturer needs to sell 50 bikes to break even.

Key Concepts

Break-even Analysis

When a company wants to know how many units it needs to sell to cover all of its costs, it conducts a break-even analysis. This is a critical financial tool used to determine the point at which a business, like Bikes'R'Us, will start to make a profit. For Bikes'R'Us, this analysis showed them exactly how many bikes they needed to sell.

The break-even point is where total revenue equals total costs, meaning there is no net profit or loss. It's also crucial for understanding the implications of pricing, cost structure and sales volume.

• Total Revenue: The total incoming money from selling goods, calculated as the selling price per unit times the number of units sold.
• Total Costs: The sum of all fixed costs (constant regardless of the number of units produced) and variable costs (which increase with each additional unit produced).

Reaching the break-even point allows a company to plan its finances effectively and identify how sensitive it is to changes in pricing or cost.

Augmented Matrix

An augmented matrix is a convenient tool for solving systems of linear equations, especially in more complex scenarios. This method represents the equations in matrix form, combining both the coefficients of the variables and the constants from the equations into one unified form.

For instance, in our Bikes'R'Us problem, we transformed the equation into an augmented matrix to make it easier to solve. By setting up an augmented matrix \[\begin{bmatrix} 70 & | & 3500 \end{bmatrix}\] we focused on the straightforward relationship between the equations' coefficients and constants.

• The coefficient represents how much the variable contributes to each equation (in this case, 70).
• The constant reflects the fixed part of the equation (here, 3500 dollars).

This method simplifies solving for the unknown variable, especially when using techniques from linear algebra.

Cost Equation

The cost equation accounts for all expenses incurred by a company to produce a certain number of products. It reflects both fixed costs, which do not change with the level of production, and variable costs, which do.

For Bikes'R'Us, the cost equation can be written as:\[180x + 3500\]where:

• \(180x\) is the variable cost component, calculated by multiplying the cost per bike by the number of bikes.
• \(3500\) is the fixed startup cost, which remains constant regardless of the number of bikes produced.

By understanding this equation, businesses can better anticipate the financial requirements necessary before achieving profitability. It's a key component in the break-even calculation, helping businesses manage their expenses more effectively.

Revenue Equation

The revenue equation expresses the total revenue a company earns from selling its products. It's generally calculated by multiplying the number of units sold by the selling price per unit.

In Bikes'R'Us' case, the revenue equation is:\[250x\]where:

• 250 represents the selling price per bike.
• \(x\) stands for the number of bikes sold.

This equation reflects how sales volume impacts financial performance. By maximizing the revenue equation, companies aim to increase profits by either selling more units or increasing the selling price. Understanding the revenue dynamics allows businesses to strategize on pricing, marketing, and production efforts effectively.