Question

A firm's weekly profit in marketing two products is given by \(P=192 x_{1}+576 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-5000\) where \(x_{1}\) and \(x_{2}\) represent the numbers of units of each product sold weekly. Estimate the average weekly profit if \(x_{1}\) varies between 40 and 50 units and \(x_{2}\) varies between 45 and 50 units.

Short answer

By following these steps, we can find the average weekly profit when \(x_{1}\) varies between 40 and 50 units and \(x_{2}\) varies between 45 and 50 units.

Step-by-step solution

Find the integrals for \(x_{1}\) and \(x_{2}\)

First, we integrate the profit function with respect to \(x_{1}\) from 40 to 50, while treating \(x_{2}\) as a constant. Then, we do the same for \(x_{2}\), from 45 to 50, treating \(x_{1}\) as a constant. We get \((192x_{1}-x_{1}^2-2x_{1}x_{2})\) and \((576x_{2}-5x_{2}^2-2x_{1}x_{2})\) respectively.

Solve the integrals and find the averages

By solving the integrals, we get the total profits over the intervals for \(x_{1}\) and \(x_{2}\). To find the average profits, we divide each total profit by the length of the interval (10 for \(x_{1}\) and 5 for \(x_{2}\)).

Substitution of values

Now we will substitute the average \(x_{1}\) and \(x_{2}\) value into the profit equation and solve to find the average profit.

Key concepts

Profit Function in Business Calculus

Profit functions are vital in understanding how a business can maximize its potential income. A profit function like the one given above, \(P=192 x_{1}+576 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-5000\), helps determine the profit based on the number of products sold. Here, \(x_{1}\) and \(x_{2}\) represent the number of units of two different products.

By substituting different values for \(x_{1}\) and \(x_{2}\) into the profit function, you can analyze how your profit changes. This can offer insights into how to adjust sales strategies to boost profitability. Business owners must understand that each term in the equation represents certain influences on profit, like sales volume (positive terms) and costs/discounts (negative terms). This understanding forms a base to make informed business decisions.

Integration's Role in Profit Analysis

Integration in calculus is a powerful tool used to calculate total quantities over an interval. When applied to a profit function, integration helps estimate total earnings over a range of sales volumes.

In the exercise, integrating with respect to \(x_{1}\) and \(x_{2}\) gives the profit accumulated over different units sold for the two products.

• Integration over \(x_{1}\) (from 40 to 50) keeps \(x_{2}\) constant, focusing purely on changes in the first product's sales.
• Similarly, integration over \(x_{2}\) (from 45 to 50) holds \(x_{1}\) constant to assess the impact of the second product's sales.

Understanding this process allows you to calculate future revenues wisely and gives a clear picture of how sales of different products contribute to the total profit.

Calculating Average Profit

The concept of average profit helps businesses evaluate their performance over a specified period or range. In our exercise, we want the average profit when \(40 \leq x_{1} \leq 50\) and \(45 \leq x_{2} \leq 50\).

To achieve this, we divide the total profit generated by the number of units in that range. This technique enables businesses to assess their profitability more realistically rather than focusing on just peaks or troughs.

This approach provides a balanced view of profitability, helping businesses understand their typical earnings and adjust strategies accordingly.

Using Marketing Analysis for Better Profits

Marketing analysis plays a critical role in improving company profits. By understanding this, business owners can make smart choices on how to market their products effectively.

When combined with profit functions and integration, marketing analysis provides insights into which aspects of their strategy are working and which are not.

• Analyzing sales data alongside the profit function allows businesses to see trends and impact of marketing efforts.
• Adjust marketing campaigns based on which product sales lead to higher profits as indicated by integration results.

This strategic approach led by data not only helps in identifying the most lucrative products but also aids in optimizing resource allocation for increased profitability.