Problem 19 - Applied Calculus

Question

A manufacturer sells two products, one at a price of \(\$ 4000\) a unit and the other at a price of \(\$ 13,000\) a unit. A quantity \(q_{1}\) of the first product and \(q_{2}\) of the second product are sold at a total cost of \(\$\left(4000+q_{1}+q_{2}\right)\) to the manufacturer. (a) Express the manufacturer's profit, \(\pi\), as a function of \(q_{1}\) and \(q_{2}\) (b) Sketch contours of \(\pi\) for \(\pi=10,000, \pi=20,000\) and \(\pi=30,000\) and the break-even curve \(\pi=0\)

Step-by-Step Solution

Answer

Profit function: \( \pi = 3999q_1 + 12999q_2 - 4000 \). Contours are found by setting different \( \pi \) values. Break-even occurs when \( \pi = 0 \).

Step 1: Define Revenue and Cost Expressions

The total revenue is the amount earned from selling the two products. Thus, the revenue \( R \) is calculated by multiplying the quantity of each product by its respective price: \( R = 4000q_1 + 13000q_2 \). The total cost \( C \) is given as \( C = 4000 + q_1 + q_2 \).

Step 2: Calculate Profit Function

Profit \( \pi \) is defined as total revenue minus total cost. Thus, the profit function for the manufacturer can be written as:\[\pi = (4000q_1 + 13000q_2) - (4000 + q_1 + q_2)\]Simplifying this, we have:\[\pi = 4000q_1 + 13000q_2 - 4000 - q_1 - q_2\]Therefore, the profit function is:\[\pi = 3999q_1 + 12999q_2 - 4000\]

Step 3: Set Profit Levels for Contours

To sketch the contour plots, solve for \( q_2 \) in terms of \( q_1 \) for each given profit level. Start with \( \pi = 10000 \):* \( 3999q_1 + 12999q_2 - 4000 = 10000 \)* Simplifying gives \( 3999q_1 + 12999q_2 = 14000 \)* \( q_2 = \frac{14000 - 3999q_1}{12999} \)Repeat the same for \( \pi = 20000 \) and \( 30000 \).

Step 4: Find Break-even Curve (\(\pi = 0\))

Set \( \pi = 0 \) in the profit function:* \( 3999q_1 + 12999q_2 - 4000 = 0 \)* Simplifying gives \( 3999q_1 + 12999q_2 = 4000 \)* \( q_2 = \frac{4000 - 3999q_1}{12999} \)This equation provides the break-even curve where there is neither profit nor loss.

Step 5: Visualizing Contours and Break-even Curve

Draw the contour plots using the equations derived in Steps 3 and 4, plotting \( q_1 \) on the x-axis and \( q_2 \) on the y-axis. The contours at \( \pi = 10000, 20000, 30000 \) and the break-even curve at \( \pi = 0 \) will show lines representing different profit levels based on different quantities of the products sold.

Key Concepts

Contour Plots

Contour plots are a useful tool in visualizing how different combinations of product quantities affect profit levels. In this context, we use contour plots to represent profit levels for a company manufacturing and selling two products. Each contour line represents a constant profit level, and the space between lines shows profit variations for different quantities of products sold.

For example, to draw a contour plot for profit levels (\(\pi\)) such as \(\pi = 10,000\), we rearrange the profit equation to find one product quantity as a function of the other:

• For example, \( q_2 = \frac{14000 - 3999q_1}{12999} \) on the contour for \(\pi = 10,000\).
• Similarly, perform the rearrangement for \(\pi = 20,000\) and \(\pi = 30,000\) to draw other contours.

These equations can then be visualized on a graph, with axes representing the quantity of each product. The contour plot helps identify how increasing or decreasing the production of either product will alter the profit level, effectively offering insights into production strategies.

Break-even Analysis

Break-even analysis is the process of determining the point at which revenue exactly equals costs, resulting in no net profit or loss (\(\pi = 0\)). In this scenario, the break-even analysis helps determine the quantity of each product needed to cover all expenses without generating a profit.

To find the break-even curve, we set the profit function to zero and solve for one product in terms of the other:

• For instance, solving \( 3999q_1 + 12999q_2 = 4000 \) provides the curve \( q_2 = \frac{4000 - 3999q_1}{12999} \).
• This equation describes all combinations of quantities \( q_1 \) and \( q_2 \) that will break-even.

Visualizing this curve on the same graph as the contour lines helps illustrate which product quantity combinations yield zero profit and identifies the changes needed to move into profitable territory.

Revenue Calculation

Revenue calculation is a fundamental concept in understanding a company's financial performance. It involves determining the total income generated from selling goods or services. In this manufacturer's case, revenue is calculated based on the quantities of two products sold and their respective selling prices.

This is achieved by multiplying the selling price of each product by the number of units sold, then summing both products' revenue:

• For product one: revenue is \( 4000q_1 \).
• For product two: revenue is \( 13000q_2 \).
• Total revenue then becomes \( R = 4000q_1 + 13000q_2 \).

Understanding how to calculate revenue gives insight into overall company earnings before deducting costs. It's essential for businesses to increase revenue by optimizing sales strategies for greater profitability.