A company produces two types of hand soaps: a solid and a liquid soap. The cost of production for a solid soap is 10 cents per soap and 20 cents for the liquid soap. The company developed a model that predicts how many units x1 (of the solid soap), and x2 (of the liquid soap) they will sell per month. According to the model, the amount of units sold depends on the prices p1,p2 (in dollars) of solid and liquid soap through the equation(equation is attached) Find the value of p1,p2 that maximizes the profit. ### Mathematical EquationsThe following equations describe a simple economic model with two variables:1. $ x_1 = 200 + 100(p_2 - p_1) $ - **Explanation**: This equation represents a linear relationship where $ x_1 $ depends on the difference between two prices, $ p_2 $ and $ p_1 $. The term $ 200 $ is the intercept, and $ 100 $ is the coefficient showing the rate of change in $ x_1 $ per unit change in the price difference.2. $ x_2 = 500 - 150p_2 $ - **Explanation**: This equation represents another linear relationship where $ x_2 $ decreases as $ p_2 $ increases. The intercept is $ 500 $, and the coefficient $ -150 $ indicates the rate at which $ x_2 $ decreases for each unit increase in $ p_2 $.### ApplicationThese equations model scenarios like consumer demand or production output based on changing prices. Understanding these relationships helps in economic analysis and decision-making.

From the given data:x1=200+100p2-p1 and x2=500-150p2Total…

Answered: A company produces two types of hand…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A company produces two types of hand soaps: a solid and a liquid soap. The cost of production for a solid soap is 10 cents per soap and 20 cents for the liquid soap. The company developed a model that predicts how many units x1 (of the solid soap), and x2 (of the liquid soap) they will sell per month. According to the model, the amount of units sold depends on the prices p1,p2 (in dollars) of solid and liquid soap through the equation

(equation is attached)

Find the value of p1,p2 that maximizes the profit.

### Mathematical Equations

The following equations describe a simple economic model with two variables:

1. $ x_1 = 200 + 100(p_2 - p_1) $

- **Explanation**: This equation represents a linear relationship where $ x_1 $ depends on the difference between two prices, $ p_2 $ and $ p_1 $. The term $ 200 $ is the intercept, and $ 100 $ is the coefficient showing the rate of change in $ x_1 $ per unit change in the price difference.

2. $ x_2 = 500 - 150p_2 $

- **Explanation**: This equation represents another linear relationship where $ x_2 $ decreases as $ p_2 $ increases. The intercept is $ 500 $, and the coefficient $ -150 $ indicates the rate at which $ x_2 $ decreases for each unit increase in $ p_2 $.

### Application

These equations model scenarios like consumer demand or production output based on changing prices. Understanding these relationships helps in economic analysis and decision-making.

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