Problem 36 - Calculus an Applied Approach

Question

After a change in marketing, the weekly profit of the firm in Exercise 35 is given by \(P=200 x_{1}+580 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-7500 .\) Estimate the average weekly profit if \(x_{1}\) varies between 55 and 65 units and \(x_{2}\) varies between 50 and 60 units.

Step-by-Step Solution

Answer

To find the average weekly profit, calculate the double integral of the profit function over the given intervals for \(x_1\) and \(x_2\) and divide this by the area of the rectangle formed by these intervals.

Step 1: Understand the given profit function

The profit function is given by \(P=200 x_{1}+580 x_{2}-x_{1}^{2}-5x_{2}^{2}-2 x_{1} x_{2} -7500\). To calculate the profit, this function needs to be evaluated at different values for \(x_1\) and \(x_2\).

Step 2: Formulate the equations for average profit

The average profit when \(x_1\) varies between 55 and 65 and \(x_2\) varies between 50 and 60 is found by integrating the profit function over these intervals and dividing by the length of the intervals. Therefore, we need to calculate the integral \(\int_{55}^{65}\int_{50}^{60} (200x_{1}+580x_{2}-x_{1}^{2}-5x_{2}^{2}-2x_{1}x_{2}-7500) dx_{1}dx_{2}\) and divide it by the area of the rectangle formed by these intervals, \((65-55) \times (60-50)\).

Step 3: Calculate the integral and the weekly profit

To calculate the integral, we need to separate it into two parts and calculate the integral with respect to \(x_1\) first, and then \(x_2\). The bounds for the integrals are given. After calculating the integral, we evaluate it at the limits to get the total profit. After calculating the double integral, we divide this by the area of the rectangle to get the average weekly profit.

Key Concepts

Calculus Integration

The concept of calculus integration is fundamental in determining the total profit generated over specific intervals. Integration helps us compute the accumulated profit over the levels of decision variables, such as quantities of product sold. In this exercise, we are given a profit function involving variables \( x_1 \) and \( x_2 \). To find the total profit when \( x_1 \) varies from 55 to 65 and \( x_2 \) varies from 50 to 60, we employ a double integral. Integration allows us to sum the continuous changes in profit across these intervals:

• Set up the double integral to add up profit contributions for small increments of \( x_1 \) and \( x_2 \).
• The function to be integrated is \( 200x_{1} + 580x_{2} - x_{1}^{2} - 5x_{2}^{2} - 2x_{1}x_{2} - 7500 \).

Double integration is done in two steps: first, integrate with respect to one variable while considering the other as a constant, and then repeat for the other variable. This process results in an expression that represents the total profit across given ranges. Calculus integration thus serves as a powerful tool to measure cumulative profit changes rather than relying on single-point calculations.

Average Profit Calculation

Average profit calculation provides a meaningful insight into the overall profitability given specific sales intervals. The average profit gives us a sense of the expected profit without needing to observe every possible profit outcome. Here is an approach to calculating the average profit:

• First, determine the total profit over the given intervals of \( x_1 \) and \( x_2 \) using calculus integration.
• Calculate the area of the region formed by these intervals, given by multiplying the range of \( x_1 \) \((65-55)\) by the range of \( x_2 \) \((60-50)\).
• Divide the integrated profit by this area to obtain the average profit.

This method ensures we account for all variations in the quantities sold within the specified ranges, providing a balanced measure of profitability. Average profit is not just a single profit number, but rather an estimation that can guide decision-making in terms of marketing and production strategies.

Multivariable Calculus

In this exercise, multivariable calculus brings us the mechanics to address problems where multiple variables impact results, such as profit. Unlike single-variable calculus, which deals with one changing quantity, multivariable calculus handles functions involving more than one variable. This profit function is a perfect example:

• It incorporates terms involving powers and products of \( x_1 \) and \( x_2 \), reflecting complex interrelationships.
• The profit is subject to changes in both \( x_1 \) and \( x_2 \), which we evaluate through partial derivatives and integration.

The double integral performed in the solution is a key aspect of multivariable calculus. It enables us to account for changes in both dimensions (\( x_1 \) and \( x_2 \)) in a two-dimensional region. This consideration is essential for calculating quantities like total and average profits. Multivariable calculus thus extends the insight from simple calculus to real-world problems with complex interdependencies.