Problem 36 - Intermediate Algebra

Question

In Problems \(34-37\) recall that the money a business spends to produce a product (or service) is called its cost and the money that it takes in from the sales of a product (or service) is called the revenue. In business and economics, it is important to determine the value at which costs equal the revenue, called the break-even point. Production Planning. A paint manufacturer can choose between two processes for making house paint, with monthly costs as shown in the table. Assume that the paint sells for \(\$ 36\) per gallon. $$ \begin{array}{|c|c|c|} \hline \text { Process } & \text { Fixed costs } & \text { Unit cost (per gallon) } \\ \hline \mathrm{A} & \$ 32,500 & \$ 26 \\ \mathrm{B} & \$ 80,600 & \mathrm{S} 10 \\ \hline \end{array} $$ a. How many gallons of paint must be sold per month for the manufacturer to break even if process \(A\) is used to produce the paint? (Hint: To break even, revenue \(=\) costs.) b. How many gallons of paint must be sold per month for the manufacturer to break even if process \(B\) is used to produce the paint? c. If expected sales are \(3,500\) gallons per month, which process should the company use?

Step-by-Step Solution

Answer

Process A break-even: 3,250 gallons; Process B break-even: 3,100 gallons; choose Process B for 3,500 gallons.

Step 1: Understanding Break-even Formula

To find the break-even point, set the Revenue equal to Costs. The revenue can be calculated as \( R = P \times Q \), where \( P \) is the price per unit and \( Q \) is the quantity. The cost equation includes fixed costs and variable costs, so it is \( C = F + V \times Q \), where \( F \) is fixed cost, and \( V \) is the variable cost per unit.

Step 2: Calculating Break-even for Process A

For Process A, use the cost equation: \( C = 32500 + 26Q \).Revenue is \( R = 36Q \). Equate them for break-even:\[ 36Q = 32500 + 26Q \]Subtract \( 26Q \) from both sides:\[ 10Q = 32500 \]Solve for \( Q \):\[ Q = \frac{32500}{10} = 3250 \] gallons.

Step 3: Calculating Break-even for Process B

For Process B, use the cost equation: \( C = 80600 + 10Q \).Revenue is \( R = 36Q \). Equate them for break-even:\[ 36Q = 80600 + 10Q \]Subtract \( 10Q \) from both sides:\[ 26Q = 80600 \]Solve for \( Q \):\[ Q = \frac{80600}{26} \approx 3100 \] gallons.

Step 4: Determining Best Process at 3500 Gallons

For 3500 gallons, calculate the total cost for each process:- **Process A**: \( C = 32500 + 26 \times 3500 = 123500 \)- **Process B**: \( C = 80600 + 10 \times 3500 = 115600 \).Find revenue: \( R = 36 \times 3500 = 126000 \).Compare costs: Process B (Cost = 115600) is cheaper than Process A (Cost = 123500).

Key Concepts

Cost and Revenue

In the world of business, understanding cost and revenue plays a crucial role in determining profitability and sustainability. **Costs** are the expenses a business incurs in the process of producing goods or providing services. These can include things like materials, labor, and overhead expenses. On the other hand, **revenue** is the money that comes into the business from selling those goods or services back to customers.

For example, if a paint manufacturer sells a gallon of paint for \( \\(36 \), that \( \\)36 \) is the revenue per unit. However, to produce each gallon of paint, there are costs involved, including raw materials and labor. The objective of a business is to ensure that the revenue exceeds the costs, leading to profit. In the exercise, we set the scenario for a paint manufacturing company, where we need to understand these two terms to solve the problem.

Fixed and Variable Costs

In business, costs are categorized into two primary types: **fixed costs** and **variable costs**.

- **Fixed Costs:** These are expenses that do not change with the level of production or sales. They remain constant regardless of how much you produce. In our paint manufacturing example, if Process A has a fixed cost of \( \\(32,500 \), this amount has to be paid every month, irrespective of whether they produce 1 gallon or 10,000 gallons of paint.- **Variable Costs:** Unlike fixed costs, variable costs change with the production level. They are the cost per unit of production. In process A, each gallon of paint costs \( \\)26 \) to produce, which is the variable cost. It means that if they produce 0 gallons, the total cost will be their fixed costs plus zero variable costs. But as production increases, these variable costs add up.

Analyzing these costs helps businesses forecast expenses more accurately and make informed decisions about pricing, budgeting, and scaling operations effectively.

Break-even Point Calculation

One of the primary goals of a business is to reach the **break-even point**. This point is crucial for decision-making, as it represents the number of units a company needs to sell to cover all of its costs. At this point, total revenue equals total costs, meaning the business is not making a profit, but it's not losing money either.

To find the break-even point, we use the formula:- **Break-even Point (in units) = Total Fixed Costs / (Price per Unit - Variable Cost per Unit)**
In our example:- For **Process A**, the break-even calculation is set using the equation \(36Q = 32500 + 26Q\), which simplifies to \(Q = 3250\) gallons. This means the manufacturer needs to sell 3250 gallons of paint to cover costs.- For **Process B**, the equation is \(36Q = 80600 + 10Q\), simplifying to \(Q \approx 3100\) gallons.

Understanding how to calculate the break-even point helps businesses assess how product decisions impact financial outcomes and informs strategic planning. For instance, if the expected sales are 3500 gallons, the manufacturer should choose Process B because it offers lower total costs, as calculated previously.