Problem 17 - Applied Calculus

Question

The table shows the cost of manufacturing various quantities of an item and the revenue obtained from their sale. $$ \begin{array}{r|r|r|r|r|r|r|r|r|r} \hline \text { Quantity } & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\ \hline \text { Cost (\$) } & 120 & 400 & 600 & 780 & 1000 & 1320 & 1800 & 2500 & 3400 \\ \hline \text { Revenue (\$) } & 0 & 300 & 600 & 900 & 1200 & 1500 & 1800 & 2100 & 2400 \\ \hline \end{array} $$ (a) What range of production levels appears to be profitable? (b) Calculate the profit or loss for each of the quantities shown. Estimate the most profitable production level.

Step-by-Step Solution

Answer

Profitable range: 50 to 60 units. Most profitable at 60 units.

Step 1: Understand Profit Calculation

Profit is calculated by subtracting the cost from revenue for each production level. Profit formula: \( \text{Profit} = \text{Revenue} - \text{Cost} \).

Step 2: Calculate Profit for Each Quantity

Use the profit formula to calculate for each provided quantity. \( \text{Quantity } 0: \text{Profit} = 0 - 120 = -120 \). Similarly, perform the calculation for each quantity to find profits.

Step 3: Identify Profitable Quantities

Examine the calculated profits for each quantity. Any quantity with a positive profit is considered profitable. Compile the list of such quantities.

Step 4: Determine Most Profitable Quantity

Compare profits for all quantities and identify the maximum profit value. The quantity corresponding to this maximum profit is the most profitable production level.

Key Concepts

Revenue and Cost Analysis

Revenue and cost analysis is an essential tool in understanding business profitability. Revenue refers to the total amount of money earned from sales, while costs are the expenses incurred to produce and sell goods. These include materials, labor, and overhead costs. Analyzing the relationship between revenue and costs helps businesses understand profitability. By evaluating how costs impact revenue, businesses can make informed decisions. To start, gather data on both revenue and costs across different production levels. In the given exercise, this data is presented in a table. Identify how revenue and cost change as production quantities increase. Compare revenue to costs to evaluate which production levels might be more financially desirable. It's important to recognize that costs don't always scale linearly with production. For instance, at higher production levels, costs may increase more dramatically due to factors like overtime pay or wear and tear on machinery. Differentiating between fixed and variable costs can be beneficial during this analysis.

Production Levels

Production levels refer to the quantity of goods produced or sold by a business. In the exercise, these are represented by quantities such as 0, 10, 20, etc., showing how outputs correlate with cost and revenue. Understanding production levels is key for designing efficient operations. By evaluating how changes in production affect both cost and revenue, businesses can make adjustments to optimize their manufacturing process and inventory management. * With low production levels, businesses might not cover fixed costs fully, resulting in losses. * As production increases, these costs are distributed over more units, potentially lowering the cost per unit. * Higher production levels usually aim to maximize profit but could lead to increased costs if not managed well. It's crucial to strike a balance where production is sufficient but not excessive, aligning the supply closely with market demand. This strategy ensures resources are used effectively without incurring unnecessary costs.

Profit Maximization

Profit maximization is the process of finding the production level where profits are at their highest. This requires assessing the difference between total revenue and total costs for various production quantities.The profit for each quantity is calculated using the formula: \[ \text{Profit} = \text{Revenue} - \text{Cost} \]In the exercise, profitable production levels are identified by calculating profits for each quantity to see where revenue exceeds the costs.To maximize profit:

• Calculate profits for each production level: Subtract costs from revenue for each level given in the table.
• Identify positive profits: Look for production quantities that yield a positive difference between revenue and cost.
• Find the maximum profit: Among the quantities with positive profits, select the one with the highest profit value as the optimal production level for maximum profitability.

Focusing on profit maximization helps companies make strategic decisions about production levels, pricing, and cost management to ensure long-term success and growth.