A Company Models Its Revenue As R ( X ) = 50 X − 2 X 2 R(x)=50x-2x^2 R ( X ) = 50 X − 2 X 2 And Expenses As E ( X ) = 10 X + 100 E(x)=10x+100 E ( X ) = 10 X + 100 . Determine The Maximum Profit.A. X = 20 X=20 X = 20 B. X = 25 X=25 X = 25 C. X = 10 X=10 X = 10 D. X = 15 X=15 X = 15

Introduction

In the world of business, understanding the relationship between revenue and expenses is crucial for making informed decisions. A company's revenue and expenses are often modeled using mathematical functions, which can be used to determine the optimal level of production or sales to maximize profit. In this article, we will explore how to model revenue and expenses using quadratic functions and determine the maximum profit.

Revenue and Expenses Modeling

The revenue of a company is often modeled using a quadratic function, which takes into account the number of units sold and the price per unit. Let's consider a company that models its revenue as $R(x)=50x-2x^2$, where $x$ represents the number of units sold. The revenue function is a quadratic function, which means it has a parabolic shape. The coefficient of $x^2$ is negative, indicating that the revenue decreases as the number of units sold increases.

The expenses of a company are also modeled using a quadratic function, which takes into account the fixed costs and the variable costs. Let's consider a company that models its expenses as $E(x)=10x+100$, where $x$ represents the number of units sold. The expenses function is a linear function, which means it has a straight line shape.

Profit Function

The profit of a company is the difference between its revenue and expenses. Let's define the profit function as $P(x)=R(x)-E(x)$. Substituting the revenue and expenses functions, we get:

$$P(x)=(50x-2x^2)-(10x+100)$$

Simplifying the profit function, we get:

$$P(x)=40x-2x^2-100$$

Finding the Maximum Profit

To find the maximum profit, we need to find the critical points of the profit function. Critical points occur when the derivative of the function is equal to zero or undefined. Let's find the derivative of the profit function:

$$P'(x)=40-4x$$

Setting the derivative equal to zero, we get:

$$40-4x=0$$

Solving for $x$, we get:

$$x=10$$

This is the critical point of the profit function. To determine whether this point corresponds to a maximum, minimum, or saddle point, we need to examine the second derivative of the profit function:

$$P''(x)=-4$$

Since the second derivative is negative, the critical point $x=10$ corresponds to a maximum.

Conclusion

In conclusion, we have modeled the revenue and expenses of a company using quadratic functions and determined the maximum profit. The maximum profit occurs when the number of units sold is 10. This result can be used by the company to determine the optimal level of production or sales to maximize profit.

Answer

The correct answer is C. $x=10$.

Discussion

This problem illustrates the importance of understanding the relationship between revenue and expenses in business. By modeling these functions using quadratic equations, companies can make informed decisions about production and sales levels to maximize profit. The critical point of the profit function corresponds to the maximum profit, which can be used to determine the optimal level of production or sales.

Additional Resources

For more information on quadratic functions and their applications in business, please refer to the following resources:

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Quadratic Equations" by Khan Academy
• [3] "Business Mathematics" by Investopedia

References

[1] Math Open Reference. (n.d.). Quadratic Functions. Retrieved from https://www.mathopenref.com/quadratic.html

[2] Khan Academy. (n.d.). Quadratic Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations

Introduction

In our previous article, we explored how to model revenue and expenses using quadratic functions and determine the maximum profit. In this article, we will answer some frequently asked questions related to maximizing profit.

Q&A

Q: What is the difference between revenue and expenses?

A: Revenue is the income generated by a company from its sales, while expenses are the costs incurred by the company to produce and sell its products or services.

Q: How do I determine the revenue function?

A: The revenue function is typically modeled using a quadratic function, which takes into account the number of units sold and the price per unit. For example, if the price per unit is $50 and the number of units sold is x, the revenue function can be modeled as R(x) = 50x - 2x^2.

Q: How do I determine the expenses function?

A: The expenses function is typically modeled using a linear function, which takes into account the fixed costs and the variable costs. For example, if the fixed costs are $100 and the variable costs are $10 per unit, the expenses function can be modeled as E(x) = 10x + 100.

Q: How do I find the maximum profit?

A: To find the maximum profit, you need to find the critical points of the profit function. Critical points occur when the derivative of the function is equal to zero or undefined. You can use calculus to find the derivative of the profit function and set it equal to zero to find the critical points.

Q: What is the significance of the second derivative in finding the maximum profit?

A: The second derivative is used to determine whether the critical point corresponds to a maximum, minimum, or saddle point. If the second derivative is negative, the critical point corresponds to a maximum.

Q: Can I use other types of functions to model revenue and expenses?

A: Yes, you can use other types of functions to model revenue and expenses, such as linear or exponential functions. However, quadratic functions are commonly used because they can model the relationship between revenue and expenses in a more accurate way.

Q: How do I apply the concept of maximizing profit in real-world business scenarios?

A: You can apply the concept of maximizing profit in real-world business scenarios by using the revenue and expenses functions to determine the optimal level of production or sales. For example, if you are a manager of a company, you can use the revenue and expenses functions to determine the optimal number of units to produce and sell to maximize profit.

Conclusion

In conclusion, maximizing profit is a crucial concept in business that can be achieved by modeling revenue and expenses using quadratic functions. By understanding the relationship between revenue and expenses, businesses can make informed decisions about production and sales levels to maximize profit.

Additional Resources

For more information on maximizing profit, please refer to the following resources:

• [1] "Maximizing Profit" by Investopedia
• [2] "Revenue and Expenses" by AccountingCoach
• [3] "Business Mathematics" by Khan Academy

References

[1] Investopedia. (n.d.). Maximizing Profit. Retrieved from https://www.investopedia.com/terms/m/maximizing-profit.asp

[2] AccountingCoach. (n.d.). Revenue and Expenses. Retrieved from https://www.accountingcoach.com/revenue-and-expenses

[3] Khan Academy. (n.d.). Business Mathematics. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations