Problem 37 - Intermediate Algebra

Question

Manufacturing. \(\quad\) A manufacturer of automobile water pumps is considering retooling for one of two manufacturing processes, with monthly fixed costs and unit costs as indicated in the table. Fach water pump can be sold for \(\$ 50\). $$ \begin{array}{|c|c|c|} \hline \text { Process } & \text { Fixed costs } & \text { Unit cost } \\ \hline \mathrm{A} & \$ 12,390 & \$ 29 \\ \mathrm{B} & \$ 20,460 & \$ 17 \\ \hline \end{array} $$ a. How many water pumps must be sold per month for the manufacturer to break even if process A is used to produce the pumps? (Hint: To break even, revenue \(=\) costs.) b. How many water pumps must be sold per month for the manufacturer to break even if process \(B\) is used to produce the pumps? (Hint: To break even, revenue \(=\) costs.) c. If expected sales are 550 water pumps per month, which process should be used?

Step-by-Step Solution

Answer

590 pumps for Process A and 620 pumps for Process B to break even. For 550 sales, Process A is better.

Step 1: Understanding Break-Even for Process A

For Process A, the fixed cost is \\(12,390 and the unit production cost is \\)29. The selling price per pump is \$50. To find the break-even point, we set the total revenue equal to the total costs. Revenue can be expressed as \(50x\) and costs as \(12,390 + 29x\), where \(x\) is the number of pumps. So, the equation is: \[50x = 12,390 + 29x\]

Step 2: Solving for Process A Break-Even Point

Solving the equation \(50x = 12,390 + 29x\), we first subtract \(29x\) from both sides: \[21x = 12,390\]. Divide both sides by 21 to solve for \(x\): \[x = \frac{12,390}{21} = 590\]. So, 590 pumps must be sold to break even using Process A.

Step 3: Understanding Break-Even for Process B

For Process B, the fixed cost is \\(20,460 and the unit production cost is \\)17. Again, the selling price per pump is \$50. Set the total revenue equal to total costs: Revenue is \(50x\), costs are \(20,460 + 17x\). The equation is: \[50x = 20,460 + 17x\]

Step 4: Solving for Process B Break-Even Point

Solving the equation \(50x = 20,460 + 17x\), subtract \(17x\) from both sides to get \[33x = 20,460\]. Divide both sides by 33 to solve for \(x\): \[x = \frac{20,460}{33} \approx 620\]. So, 620 pumps must be sold to break even using Process B.

Step 5: Comparison for 550 Pump Sales

If the company expects to sell 550 pumps, calculate the profit for each process. For Process A, the cost is \(12,390 + 29 \times 550 = 28,340\) and revenue is \(50 \times 550 = 27,500\), giving a loss of \(27,500 - 28,340 = -840\). For Process B, the cost is \(20,460 + 17 \times 550 = 29,810\), giving a loss of \(27,500 - 29,810 = -2,310\). Process A yields a smaller loss and is preferable.

Key Concepts

Fixed Costs

In any manufacturing business, fixed costs are expenses that remain the same, regardless of the number of goods produced. These costs are crucial as they form the backbone of a company’s budget, impacting financial planning and decision-making. They include costs such as rent, salaries, and equipment depreciation. For our exercise, Process A has a fixed cost of \(\\(12,390\), while Process B's is higher at \(\\)20,460\).

Understanding fixed costs is essential for break-even analysis, which determines how much product needs to be sold to cover both fixed and variable costs. Fixed costs do not fluctuate with production volume, meaning they must be covered even if the production quantity is zero. This characteristic differentiates them from variable costs. In break-even analysis, fixed costs significantly influence the number of units needed to be sold in order to not incur a loss.

Variable Costs

Variable costs are expenses that change directly with the level of production output. They include costs related to materials and direct production labor. Unlike fixed costs, variable costs increase with each additional unit produced, making them essential for calculating total production costs. In our example, Process A has a unit cost of \(\\(29\) per pump, while Process B, despite higher fixed costs, offers a lower unit cost of \(\\)17\) per pump.

Knowing the variable costs is critical because they directly impact the profit margins of each product sold. If variable costs are too high relative to the selling price, they can significantly eat into profits or even lead to losses if sales fall short of expectations. Hence, companies often strive to balance fixed costs and variable costs to optimize revenue and maintain profitability over time.

Revenue Analysis

Revenue analysis involves studying the income generated from selling goods or services. It's an essential component of business strategy as it helps in determining the financial viability of different production processes. To perform a revenue analysis for the manufacturing of water pumps, we consider the selling price of \(\$50\) per pump.

In our exercise, revenue for any number of pumps can be expressed mathematically as \(50x\), where \(x\) is the number of pumps sold. The goal of revenue analysis is to find out how many units need to be sold to not only cover both fixed and variable costs but also to begin generating profit. In the context of this exercise, conducting a thorough revenue analysis allowed us to calculate how many units of water pumps must be sold at a minimum to break even in both processes (590 pumps for Process A and 620 pumps for Process B).

Ultimately, revenue analysis helps in choosing between different manufacturing processes by examining their financial implications, especially when expected sales fluctuate, as in our example.