The Revenue From Selling $x$ Shirts Is $r(x)=15x$.The Cost Of Buying $ X X X [/tex] Shirts Is $c(x)=7x+20$.The Profit From Selling $x$ Shirts Is $p(x)=r(x)-c(x)$.What Is

Understanding the Problem

In this problem, we are given the revenue function $r(x)$, which represents the revenue from selling $x$ shirts, and the cost function $c(x)$, which represents the cost of buying $x$ shirts. We are also asked to find the profit function $p(x)$, which represents the profit from selling $x$ shirts.

Revenue Function

The revenue function $r(x)$ is given by the equation $r(x) = 15x$. This means that for every shirt sold, the revenue increases by $15. For example, if $x = 10$, the revenue would be $r(10) = 15 \times 10 = 150$.

Cost Function

The cost function $c(x)$ is given by the equation $c(x) = 7x + 20$. This means that for every shirt bought, the cost increases by $7, and there is an additional fixed cost of $20. For example, if $x = 10$, the cost would be $c(10) = 7 \times 10 + 20 = 90$.

Profit Function

The profit function $p(x)$ is given by the equation $p(x) = r(x) - c(x)$. This means that the profit is the difference between the revenue and the cost. For example, if $x = 10$, the profit would be $p(10) = r(10) - c(10) = 150 - 90 = 60$.

Calculating the Profit

To calculate the profit, we need to subtract the cost from the revenue. This can be done using the equation $p(x) = r(x) - c(x)$. We can substitute the given equations for $r(x)$ and $c(x)$ to get:

$$p(x) = 15x - (7x + 20)$$

Simplifying the equation, we get:

$$p(x) = 15x - 7x - 20$$

$$p(x) = 8x - 20$$

This means that the profit from selling $x$ shirts is $8x - 20$.

Graphing the Profit Function

To graph the profit function, we can use a graphing calculator or a computer program. The graph of the profit function would be a straight line with a slope of 8 and a y-intercept of -20.

Interpreting the Graph

The graph of the profit function can be interpreted as follows:

• The x-axis represents the number of shirts sold.
• The y-axis represents the profit.
• The slope of the line represents the rate of change of the profit with respect to the number of shirts sold.
• The y-intercept represents the profit when no shirts are sold.

Conclusion

In this problem, we have calculated the revenue, cost, and profit functions for selling shirts. We have also graphed the profit function and interpreted the graph. The profit function is a straight line with a slope of 8 and a y-intercept of -20. This means that the profit from selling $x$ shirts is $8x - 20$.

Applications of the Profit Function

The profit function can be used in a variety of applications, such as:

• Business planning: The profit function can be used to determine the optimal number of shirts to sell in order to maximize profit.
• Marketing: The profit function can be used to determine the most effective marketing strategies in order to increase revenue and profit.
• Financial analysis: The profit function can be used to analyze the financial performance of a business and make informed decisions about investments and resource allocation.

Limitations of the Profit Function

The profit function has several limitations, including:

• Assumes linear relationships: The profit function assumes that the relationships between revenue, cost, and profit are linear. However, in reality, these relationships may be non-linear.
• Does not account for external factors: The profit function does not account for external factors such as changes in market demand, competition, and economic conditions.
• Requires accurate data: The profit function requires accurate data on revenue, cost, and profit. However, this data may be difficult to obtain or may be subject to errors.

Future Research Directions

Future research directions for the profit function include:

• Non-linear relationships: Investigating non-linear relationships between revenue, cost, and profit.
• External factors: Accounting for external factors such as changes in market demand, competition, and economic conditions.
• Data quality: Improving the accuracy and reliability of data on revenue, cost, and profit.

Conclusion

Q: What is the revenue function?

A: The revenue function is a mathematical equation that represents the revenue from selling a certain number of shirts. In this case, the revenue function is given by the equation $r(x) = 15x$, where $x$ is the number of shirts sold.

Q: What is the cost function?

A: The cost function is a mathematical equation that represents the cost of buying a certain number of shirts. In this case, the cost function is given by the equation $c(x) = 7x + 20$, where $x$ is the number of shirts bought.

Q: What is the profit function?

A: The profit function is a mathematical equation that represents the profit from selling a certain number of shirts. 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.

Q: How do I calculate the profit?

A: To calculate the profit, you need to subtract the cost from the revenue. This can be done using the equation $p(x) = r(x) - c(x)$. You can substitute the given equations for $r(x)$ and $c(x)$ to get:

$$p(x) = 15x - (7x + 20)$$

Simplifying the equation, you get:

$$p(x) = 15x - 7x - 20$$

$$p(x) = 8x - 20$$

This means that the profit from selling $x$ shirts is $8x - 20$.

Q: What is the graph of the profit function?

A: The graph of the profit function is a straight line with a slope of 8 and a y-intercept of -20.

Q: What does the graph of the profit function represent?

A: The graph of the profit function represents the relationship between the number of shirts sold and the profit. The x-axis represents the number of shirts sold, and the y-axis represents the profit.

Q: What are some applications of the profit function?

A: The profit function can be used in a variety of applications, such as:

• Business planning: The profit function can be used to determine the optimal number of shirts to sell in order to maximize profit.
• Marketing: The profit function can be used to determine the most effective marketing strategies in order to increase revenue and profit.
• Financial analysis: The profit function can be used to analyze the financial performance of a business and make informed decisions about investments and resource allocation.

Q: What are some limitations of the profit function?

A: The profit function has several limitations, including:

• Assumes linear relationships: The profit function assumes that the relationships between revenue, cost, and profit are linear. However, in reality, these relationships may be non-linear.
• Does not account for external factors: The profit function does not account for external factors such as changes in market demand, competition, and economic conditions.
• Requires accurate data: The profit function requires accurate data on revenue, cost, and profit. However, this data may be difficult to obtain or may be subject to errors.

Q: What are some future research directions for the profit function?

A: Some future research directions for the profit function include:

• Non-linear relationships: Investigating non-linear relationships between revenue, cost, and profit.
• External factors: Accounting for external factors such as changes in market demand, competition, and economic conditions.
• Data quality: Improving the accuracy and reliability of data on revenue, cost, and profit.

Q: How can I use the profit function in real-world applications?

A: The profit function can be used in a variety of real-world applications, such as:

• Business planning: Use the profit function to determine the optimal number of shirts to sell in order to maximize profit.
• Marketing: Use the profit function to determine the most effective marketing strategies in order to increase revenue and profit.
• Financial analysis: Use the profit function to analyze the financial performance of a business and make informed decisions about investments and resource allocation.

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

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

• Assuming linear relationships: Do not assume that the relationships between revenue, cost, and profit are linear. Instead, investigate non-linear relationships.
• Failing to account for external factors: Do not fail to account for external factors such as changes in market demand, competition, and economic conditions.
• Using inaccurate data: Do not use inaccurate data on revenue, cost, and profit. Instead, use accurate and reliable data.

Q: How can I improve the accuracy and reliability of the profit function?

A: To improve the accuracy and reliability of the profit function, you can:

• Use accurate data: Use accurate and reliable data on revenue, cost, and profit.
• Investigate non-linear relationships: Investigate non-linear relationships between revenue, cost, and profit.
• Account for external factors: Account for external factors such as changes in market demand, competition, and economic conditions.