The Profit Earned By A Hot Dog Stand Is A Linear Function Of The Number Of Hot Dogs Sold. It Costs The Owner $ 48 \$48 $48 Each Morning For The Day's Supply Of Hot Dogs, Buns, And Mustard, But He Earns $ 2 \$2 $2 Profit For Each Hot Dog Sold.

Mar 9, 2025 by ADMIN 243 views

**The Profit Earned by a Hot Dog Stand: A Linear Function of the Number of Hot Dogs Sold**

Introduction

A hot dog stand is a common sight at many outdoor events, festivals, and even in busy city streets. The owner of the hot dog stand earns a profit from selling hot dogs, but the profit is not constant and depends on the number of hot dogs sold. In this article, we will explore how the profit earned by a hot dog stand is a linear function of the number of hot dogs sold.

The Cost of Running the Hot Dog Stand

The owner of the hot dog stand incurs a cost of $\$48$ each morning for the day's supply of hot dogs, buns, and mustard. This cost is a fixed expense and does not depend on the number of hot dogs sold. The cost is a one-time expense, and the owner does not earn any profit from it.

The Profit Earned from Selling Hot Dogs

The owner earns a profit of $\$2$ for each hot dog sold. This profit is a variable expense, and it depends on the number of hot dogs sold. The more hot dogs sold, the more profit the owner earns.

The Linear Function

The profit earned by the hot dog stand is a linear function of the number of hot dogs sold. A linear function is a function that can be written in the form $f(x) = mx + b$ , where $m$ is the slope of the function and $b$ is the y-intercept.

In this case, the profit function can be written as $P(x) = 2x - 48$ , where $P(x)$ is the profit earned and $x$ is the number of hot dogs sold. The slope of the function is $2$ , which means that the profit increases by $\$2$ for each hot dog sold. The y-intercept is $-48$ , which means that the owner incurs a cost of $\$48$ each morning, even if no hot dogs are sold.

Graphing the Linear Function

The linear function can be graphed on a coordinate plane. The x-axis represents the number of hot dogs sold, and the y-axis represents the profit earned.

import matplotlib.pyplot as plt
import numpy as np
def P(x):
return 2*x - 48

x = np.linspace(0, 100, 100)

y = P(x)

plt.plot(x, y)
plt.xlabel('Number of Hot Dogs Sold')
plt.ylabel('Profit Earned')
plt.title('Profit Earned by a Hot Dog Stand')
plt.grid(True)
plt.show()

Interpreting the Graph

The graph of the linear function shows that the profit earned by the hot dog stand increases as the number of hot dogs sold increases. The graph is a straight line with a positive slope, indicating that the profit increases by $\$2$ for each hot dog sold.

Solving for the Number of Hot Dogs Sold

The linear function can be used to solve for the number of hot dogs sold that will result in a certain profit. For example, if the owner wants to earn a profit of $\$100$ , we can set up the equation $2x - 48 = 100$ and solve for $x$ .

# Define the equation
def equation(x):
    return 2*x - 48 - 100

from scipy.optimize import fsolve
x = fsolve(equation, 0)

print('The number of hot dogs sold is:', x)

Conclusion

In conclusion, the profit earned by a hot dog stand is a linear function of the number of hot dogs sold. The linear function can be written as $P(x) = 2x - 48$ , where $P(x)$ is the profit earned and $x$ is the number of hot dogs sold. The graph of the linear function shows that the profit increases as the number of hot dogs sold increases. The linear function can be used to solve for the number of hot dogs sold that will result in a certain profit.

References

[1] Khan Academy. (n.d.). Linear Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f-linear-functions/x2f1f-1-linear-functions-intro/v/linear-functions-intro
[2] Math Is Fun. (n.d.). Linear Functions. Retrieved from https://www.mathisfun.com/algebra/linear-functions.html

Appendix

The following is a list of formulas and equations used in this article:

$P(x) = 2x - 48$
$P(x) = mx + b$
$m = 2$
$b = -48$
$x = \frac{P(x) + 48}{2}$
The Profit Earned by a Hot Dog Stand: A Linear Function of the Number of Hot Dogs Sold - Q&A =====================================================================================

Introduction

In our previous article, we explored how the profit earned by a hot dog stand is a linear function of the number of hot dogs sold. We discussed the cost of running the hot dog stand, the profit earned from selling hot dogs, and the linear function that represents the profit earned. In this article, we will answer some frequently asked questions about the profit earned by a hot dog stand.

Q: What is the cost of running the hot dog stand?

A: The cost of running the hot dog stand is $\$48$ each morning for the day's supply of hot dogs, buns, and mustard.

Q: How much profit does the owner earn for each hot dog sold?

A: The owner earns a profit of $\$2$ for each hot dog sold.

Q: What is the linear function that represents the profit earned?

A: The linear function that represents the profit earned is $P(x) = 2x - 48$ , where $P(x)$ is the profit earned and $x$ is the number of hot dogs sold.

Q: How can I graph the linear function?

A: You can graph the linear function using a coordinate plane. The x-axis represents the number of hot dogs sold, and the y-axis represents the profit earned.

import matplotlib.pyplot as plt
import numpy as np

def P(x):
return 2*x - 48

x = np.linspace(0, 100, 100)

y = P(x)

plt.plot(x, y)
plt.xlabel('Number of Hot Dogs Sold')
plt.ylabel('Profit Earned')
plt.title('Profit Earned by a Hot Dog Stand')
plt.grid(True)
plt.show()

Q: How can I solve for the number of hot dogs sold that will result in a certain profit?

A: You can solve for the number of hot dogs sold that will result in a certain profit by setting up the equation $2x - 48 = P(x)$ and solving for $x$ .

# Define the equation
def equation(x):
    return 2*x - 48 - 100

from scipy.optimize import fsolve
x = fsolve(equation, 0)

print('The number of hot dogs sold is:', x)

Q: What is the slope of the linear function?

A: The slope of the linear function is $2$ , which means that the profit increases by $\$2$ for each hot dog sold.

Q: What is the y-intercept of the linear function?

A: The y-intercept of the linear function is $-48$ , which means that the owner incurs a cost of $\$48$ each morning, even if no hot dogs are sold.

Q: Can I use the linear function to predict the profit earned by the hot dog stand?

A: Yes, you can use the linear function to predict the profit earned by the hot dog stand. Simply plug in the number of hot dogs sold into the linear function and solve for the profit earned.

Conclusion

In conclusion, the profit earned by a hot dog stand is a linear function of the number of hot dogs sold. The linear function can be used to solve for the number of hot dogs sold that will result in a certain profit, and it can be graphed on a coordinate plane. We hope that this article has been helpful in answering your questions about the profit earned by a hot dog stand.

References

[1] Khan Academy. (n.d.). Linear Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f-linear-functions/x2f1f-1-linear-functions-intro/v/linear-functions-intro
[2] Math Is Fun. (n.d.). Linear Functions. Retrieved from https://www.mathisfun.com/algebra/linear-functions.html

Appendix

The following is a list of formulas and equations used in this article:

$P(x) = 2x - 48$
$P(x) = mx + b$
$m = 2$
$b = -48$
$x = \frac{P(x) + 48}{2}$