The Price Received For A Bicycle Is Given By The Equation $b = 100 - 10x^2$, Where $x$ Is The Number Of Bicycles Produced, In Millions.It Costs The Company $60 To Make Each Bicycle. What Is The Profit Function For The Company?A.

Introduction

In the world of business, understanding the relationship between the price received for a product and the number of units produced is crucial for making informed decisions. The given equation, $b = 100 - 10x^2$, represents the price received for a bicycle, where $x$ is the number of bicycles produced, in millions. This equation is a quadratic function that describes the price as a function of the number of bicycles produced. In this article, we will explore the concept of profit and derive the profit function for the company.

Understanding the Cost and Revenue Functions

Before we can derive the profit function, we need to understand the cost and revenue functions. The cost function represents the total cost of producing a certain number of units, while the revenue function represents the total revenue generated from selling a certain number of units.

Cost Function

The cost function is given by the equation $C(x) = 60x$, where $x$ is the number of bicycles produced, in millions. This equation represents the total cost of producing $x$ million bicycles, where each bicycle costs $60 to make.

Revenue Function

The revenue function is given by the equation $R(x) = bx$, where $b$ is the price received for each bicycle and $x$ is the number of bicycles produced, in millions. Using the given equation $b = 100 - 10x^2$, we can substitute $b$ into the revenue function to get:

$$R(x) = (100 - 10x^2)x$$

Simplifying the Revenue Function

To simplify the revenue function, we can expand the equation:

$$R(x) = 100x - 10x^3$$

Profit Function

The profit function is given by the equation $P(x) = R(x) - C(x)$, where $R(x)$ is the revenue function and $C(x)$ is the cost function. Substituting the revenue function and cost function, we get:

$$P(x) = (100x - 10x^3) - 60x$$

Simplifying the Profit Function

To simplify the profit function, we can combine like terms:

$$P(x) = 100x - 10x^3 - 60x$$

$$P(x) = 40x - 10x^3$$

Graphing the Profit Function

To visualize the profit function, we can graph the equation $P(x) = 40x - 10x^3$. The graph will show the profit as a function of the number of bicycles produced.

Interpreting the Graph

The graph of the profit function will show a parabola that opens downward. This means that the profit will initially increase as the number of bicycles produced increases, but will eventually decrease as the number of bicycles produced continues to increase.

Maximizing Profit

To maximize profit, we need to find the value of $x$ that maximizes the profit function. This can be done by taking the derivative of the profit function and setting it equal to zero:

$$\frac{dP}{dx} = 40 - 30x^2 = 0$$

Solving for $x$, we get:

$$x = \pm \sqrt{\frac{40}{30}}$$

Since $x$ represents the number of bicycles produced, it must be a positive value. Therefore, we take the positive square root:

$$x = \sqrt{\frac{40}{30}}$$

Conclusion

In conclusion, the profit function for the company is given by the equation $P(x) = 40x - 10x^3$. This equation represents the profit as a function of the number of bicycles produced. To maximize profit, we need to find the value of $x$ that maximizes the profit function. This can be done by taking the derivative of the profit function and setting it equal to zero.

References

• [1] "Profit Maximization" by Investopedia
• [2] "Cost and Revenue Functions" by Khan Academy

Further Reading

• "Profit Maximization" by Coursera
• "Cost and Revenue Functions" by edX

Appendix

• Derivative of the Profit Function
  • The derivative of the profit function is given by the equation $\frac{dP}{dx} = 40 - 30x^2$.
  • To find the maximum profit, we set the derivative equal to zero and solve for $x$.
• Solving for $x$
  • We solve for $x$ by taking the square root of both sides of the equation:
    • $x = \pm \sqrt{\frac{40}{30}}$
    • Since $x$ represents the number of bicycles produced, it must be a positive value. Therefore, we take the positive square root:
      • $x = \sqrt{\frac{40}{30}}$
        The Price Received for a Bicycle and the Profit Function: Q&A ===========================================================

Introduction

In our previous article, we explored the concept of profit and derived the profit function for a company that produces bicycles. The profit function is given by the equation $P(x) = 40x - 10x^3$, where $x$ is the number of bicycles produced, in millions. In this article, we will answer some frequently asked questions about the profit function and provide additional insights into the world of profit maximization.

Q: What is the profit function, and how is it related to the number of bicycles produced?

A: The profit function is a mathematical equation that represents the profit as a function of the number of bicycles produced. In this case, the profit function is given by the equation $P(x) = 40x - 10x^3$, where $x$ is the number of bicycles produced, in millions.

Q: How do I maximize profit using the profit function?

A: To maximize profit, we need to find the value of $x$ that maximizes the profit function. This can be done by taking the derivative of the profit function and setting it equal to zero. The derivative of the profit function is given by the equation $\frac{dP}{dx} = 40 - 30x^2$. Setting this equal to zero and solving for $x$, we get:

$$x = \sqrt{\frac{40}{30}}$$

Q: What is the significance of the value of $x$ that maximizes profit?

A: The value of $x$ that maximizes profit represents the optimal number of bicycles that should be produced to maximize profit. In this case, the optimal number of bicycles is given by the equation $x = \sqrt{\frac{40}{30}}$.

Q: How do I interpret the graph of the profit function?

A: The graph of the profit function is a parabola that opens downward. This means that the profit will initially increase as the number of bicycles produced increases, but will eventually decrease as the number of bicycles produced continues to increase.

Q: What are some common mistakes to avoid when working with the profit function?

A: Some common mistakes to avoid when working with the profit function include:

• Not taking the derivative of the profit function correctly
• Not setting the derivative equal to zero to find the maximum profit
• Not solving for $x$ correctly
• Not interpreting the graph of the profit function correctly

Q: How can I apply the concepts of profit maximization to real-world business scenarios?

A: The concepts of profit maximization can be applied to real-world business scenarios in a variety of ways. For example, a company may use the profit function to determine the optimal number of products to produce, or to determine the optimal price to charge for a product.

Q: What are some additional resources for learning more about profit maximization?

A: Some additional resources for learning more about profit maximization include:

• "Profit Maximization" by Investopedia
• "Cost and Revenue Functions" by Khan Academy
• "Profit Maximization" by Coursera
• "Cost and Revenue Functions" by edX

Conclusion

In conclusion, the profit function is a powerful tool for maximizing profit in business. By understanding the concepts of profit maximization and applying them to real-world business scenarios, companies can make informed decisions and achieve their goals. We hope that this article has provided you with a better understanding of the profit function and how it can be used to maximize profit.

References

• [1] "Profit Maximization" by Investopedia
• [2] "Cost and Revenue Functions" by Khan Academy
• [3] "Profit Maximization" by Coursera
• [4] "Cost and Revenue Functions" by edX

Further Reading

• "Profit Maximization" by Harvard Business Review
• "Cost and Revenue Functions" by MIT OpenCourseWare
• "Profit Maximization" by Stanford University
• "Cost and Revenue Functions" by University of California, Berkeley

Appendix

• Derivative of the Profit Function
  • The derivative of the profit function is given by the equation $\frac{dP}{dx} = 40 - 30x^2$.
  • To find the maximum profit, we set the derivative equal to zero and solve for $x$.
• Solving for $x$
  • We solve for $x$ by taking the square root of both sides of the equation:
    • $x = \pm \sqrt{\frac{40}{30}}$
    • Since $x$ represents the number of bicycles produced, it must be a positive value. Therefore, we take the positive square root:
      • $x = \sqrt{\frac{40}{30}}$