Problem 26 - Applied Calculus

Question

The table shows the cost of manufacturing various quantities of an item and the revenue obtained from their sale. $$\begin{array}{r|r|r|r|r|r|r|r|r|r}\hline \text { Quantity } & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\ \hline \text { cost (S) } & 120 & 400 & 600 & 780 & 1000 & 1320 & 1800 & 2500 & 3400 \\\\\hline \text { Revenue (S) } & 0 & 300 & 600 & 900 & 1200 & 1500 & 1800 & 2100 & 2400 \\\\\hline \end{array}$$ (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 quantities are 30, 40, and 50. Most profitable is at 40 with a profit of 200.

Step 1: Understand the Problem

We need to determine which production levels are profitable and calculate profit or loss for each quantity. Profit is the difference between revenue and cost.

Step 2: Calculate Profit or Loss for Each Quantity

To find profit or loss, subtract the cost from the revenue at each quantity level. Use the formula: \ \( \text{Profit} = \text{Revenue} - \text{Cost} \) \ Perform this calculation for each quantity (0, 10, 20, ..., 80):- For 0: \( 0 - 120 = -120 \)- For 10: \( 300 - 400 = -100 \)- For 20: \( 600 - 600 = 0 \)- For 30: \( 900 - 780 = 120 \)- For 40: \( 1200 - 1000 = 200 \)- For 50: \( 1500 - 1320 = 180 \)- For 60: \( 1800 - 1800 = 0 \)- For 70: \( 2100 - 2500 = -400 \)- For 80: \( 2400 - 3400 = -1000 \).

Step 3: Identify Profitable Quantities

A production level is profitable if the calculated profit is greater than zero. From Step 2, the profitable quantities are: - 30 (Profit = 120) - 40 (Profit = 200) - 50 (Profit = 180) These quantities yield a positive profit.

Step 4: Determine the Most Profitable Production Level

Compare the profits from the profitable quantities (30, 40, 50) to identify the maximum profit. - Quantity 30: Profit = 120 - Quantity 40: Profit = 200 - Quantity 50: Profit = 180 The highest profit is at a production level of 40, with a profit of 200.

Key Concepts

Revenue Calculation

Revenue calculation is a fundamental aspect of profit analysis. It represents the total income generated from selling goods or services. To compute the revenue for any quantity, use the formula: \( \text{Revenue} = \text{Price per Item} \times \text{Quantity Sold} \).
In our context, the revenue values were directly given for different quantities. They reflect how much money the business makes at each level of production.
Understanding revenue is crucial because it helps identify the sales performance at different output levels. By comparing revenue against costs, businesses can assess profitability.
Higher revenue generally indicates better sales performance, but it doesn't always mean higher profit, as costs must also be considered.

Cost Analysis

Cost analysis involves examining the costs associated with producing goods or services. Costs can include materials, labor, utilities, and other expenses.
For calculating costs in our exercise, the costs were specified for each production quantity. Knowing the cost at different production levels helps businesses make informed decisions to manage expenses efficiently.
This analysis is essential because controlling costs can directly impact profitability. Here's a quick list of consideration when analyzing costs:

• Fixed Costs: Expenses that remain constant regardless of output.
• Variable Costs: Costs that change with the level of production.
• Total Costs: The sum of fixed and variable costs at each production level.

In the exercise, identifying the costs for each quantity level allows for a precise profit or loss calculation.

Profit Maximization

Profit maximization aims to identify the output level that yields the highest possible profit. It is achieved by balancing revenue and costs effectively.
In the provided exercise, profit is calculated by subtracting the cost from revenue for each production quantity:
\[ \text{Profit} = \text{Revenue} - \text{Cost} \]
To find the most profitable production level, compare the profits calculated for each quantity:

• Profit at 30 units = 120
• Profit at 40 units = 200
• Profit at 50 units = 180

The maximum profit is at a production level of 40 units, as it yields the highest profit of 200.
The goal of profit maximization is to use this analysis to decide the optimal production strategy that ensures the company's financial health and sustainability.