The Price Charged For A Candy Bar Is P ( X P(x P ( X ] Cents, Where X X X Thousand Candy Bars Will Be Sold In A Certain City. The Price Function Is Given By P ( X ) = 62 − X 12 P(x)=62-\frac{x}{12} P ( X ) = 62 − 12 X ​ . How Many Candy Bars Must Be Sold To Maximize Revenue?

Introduction

In the world of economics and business, understanding the relationship between price and revenue is crucial for making informed decisions. The price function, which describes the relationship between the price of a product and the quantity sold, is a fundamental concept in economics. In this article, we will explore how to maximize revenue using a candy bar price function.

The Price Function

The price function for a candy bar is given by $p(x)=62-\frac{x}{12}$, where $x$ thousand candy bars will be sold in a certain city. The price function is a linear function, which means that it has a constant rate of change. The graph of the price function is a straight line with a negative slope, indicating that as the quantity sold increases, the price decreases.

Revenue Function

The revenue function, which describes the total amount of money earned from selling a certain quantity of candy bars, is given by $R(x)=xp(x)$. To find the revenue function, we multiply the price function by the quantity sold. Substituting the price function into the revenue function, we get:

$$R(x)=x\left(62-\frac{x}{12}\right)$$

Expanding the Revenue Function

To find the maximum revenue, we need to find the critical points of the revenue function. To do this, we need to expand the revenue function. Expanding the revenue function, we get:

$$R(x)=62x-\frac{x^2}{12}$$

Finding the Critical Points

To find the critical points of the revenue function, we need to take the derivative of the revenue function and set it equal to zero. Taking the derivative of the revenue function, we get:

$$R'(x)=62-\frac{x}{6}$$

Setting the derivative equal to zero, we get:

$$62-\frac{x}{6}=0$$

Solving for $x$, we get:

$$x=372$$

Interpreting the Results

The critical point $x=372$ represents the quantity of candy bars that must be sold to maximize revenue. Since $x$ represents the number of thousand candy bars sold, the quantity of candy bars that must be sold to maximize revenue is 372,000.

Conclusion

In conclusion, the price function $p(x)=62-\frac{x}{12}$ describes the relationship between the price of a candy bar and the quantity sold. The revenue function $R(x)=xp(x)$ describes the total amount of money earned from selling a certain quantity of candy bars. By finding the critical points of the revenue function, we can determine the quantity of candy bars that must be sold to maximize revenue. In this case, the quantity of candy bars that must be sold to maximize revenue is 372,000.

Maximizing Revenue with a Candy Bar Price Function: A Step-by-Step Guide

Step 1: Define the Price Function

The price function for a candy bar is given by $p(x)=62-\frac{x}{12}$, where $x$ thousand candy bars will be sold in a certain city.

Step 2: Define the Revenue Function

The revenue function, which describes the total amount of money earned from selling a certain quantity of candy bars, is given by $R(x)=xp(x)$.

Step 3: Expand the Revenue Function

To find the maximum revenue, we need to find the critical points of the revenue function. To do this, we need to expand the revenue function. Expanding the revenue function, we get:

$$R(x)=62x-\frac{x^2}{12}$$

Step 4: Find the Critical Points

To find the critical points of the revenue function, we need to take the derivative of the revenue function and set it equal to zero. Taking the derivative of the revenue function, we get:

$$R'(x)=62-\frac{x}{6}$$

Setting the derivative equal to zero, we get:

$$62-\frac{x}{6}=0$$

Solving for $x$, we get:

$$x=372$$

Step 5: Interpret the Results

The critical point $x=372$ represents the quantity of candy bars that must be sold to maximize revenue. Since $x$ represents the number of thousand candy bars sold, the quantity of candy bars that must be sold to maximize revenue is 372,000.

Maximizing Revenue with a Candy Bar Price Function: Frequently Asked Questions

Q: What is the price function for a candy bar?

A: The price function for a candy bar is given by $p(x)=62-\frac{x}{12}$, where $x$ thousand candy bars will be sold in a certain city.

Q: What is the revenue function for a candy bar?

A: The revenue function, which describes the total amount of money earned from selling a certain quantity of candy bars, is given by $R(x)=xp(x)$.

Q: How do I find the maximum revenue?

A: To find the maximum revenue, you need to find the critical points of the revenue function. To do this, you need to take the derivative of the revenue function and set it equal to zero.

Q: What is the quantity of candy bars that must be sold to maximize revenue?

A: The quantity of candy bars that must be sold to maximize revenue is 372,000.
Maximizing Revenue with a Candy Bar Price Function: A Q&A Article

Q: What is the price function for a candy bar?

A: The price function for a candy bar is given by $p(x)=62-\frac{x}{12}$, where $x$ thousand candy bars will be sold in a certain city. This function describes the relationship between the price of a candy bar and the quantity sold.

Q: What is the revenue function for a candy bar?

A: The revenue function, which describes the total amount of money earned from selling a certain quantity of candy bars, is given by $R(x)=xp(x)$. This function is a key concept in economics and business, as it helps businesses understand how to maximize their revenue.

Q: How do I find the maximum revenue?

A: To find the maximum revenue, you need to find the critical points of the revenue function. To do this, you need to take the derivative of the revenue function and set it equal to zero. This will give you the quantity of candy bars that must be sold to maximize revenue.

Q: What is the derivative of the revenue function?

A: The derivative of the revenue function is given by $R'(x)=62-\frac{x}{6}$. This derivative represents the rate of change of the revenue function with respect to the quantity sold.

Q: How do I set the derivative equal to zero?

A: To set the derivative equal to zero, you need to solve the equation $62-\frac{x}{6}=0$. This will give you the critical point of the revenue function, which represents the quantity of candy bars that must be sold to maximize revenue.

Q: What is the critical point of the revenue function?

A: The critical point of the revenue function is given by $x=372$. This represents the quantity of candy bars that must be sold to maximize revenue.

Q: What is the quantity of candy bars that must be sold to maximize revenue?

A: The quantity of candy bars that must be sold to maximize revenue is 372,000. This is the key takeaway from our analysis of the candy bar price function.

Q: How does the candy bar price function relate to real-world business decisions?

A: The candy bar price function is a simplified example of a real-world business decision. In reality, businesses must consider many factors when setting prices, including production costs, market demand, and competition. However, the principles of maximizing revenue using a price function remain the same.

Q: Can I apply the candy bar price function to other business scenarios?

A: Yes, the candy bar price function can be applied to other business scenarios, such as maximizing revenue from selling a product or service. The key is to understand the relationship between the price and quantity sold, and to use this information to make informed business decisions.

Q: What are some common mistakes to avoid when using a price function to maximize revenue?

A: Some common mistakes to avoid when using a price function to maximize revenue include:

• Not considering the relationship between price and quantity sold
• Not taking into account production costs and other expenses
• Not considering market demand and competition
• Not using a realistic price function that reflects real-world business scenarios

Q: How can I use a price function to maximize revenue in a real-world business scenario?

A: To use a price function to maximize revenue in a real-world business scenario, you need to:

• Define the price function based on the business scenario
• Calculate the revenue function
• Find the critical points of the revenue function
• Use the critical points to make informed business decisions

By following these steps, you can use a price function to maximize revenue in a real-world business scenario.