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

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Introduction

A hot dog stand is a small business that sells hot dogs to customers. The profit earned by the hot dog stand is directly related to the number of hot dogs sold. In this article, we will explore the relationship between the number of hot dogs sold and the profit earned by the hot dog stand. We will use a linear function to model this relationship and determine the equation that represents the profit earned by the hot dog stand.

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 that the owner must pay regardless of the number of hot dogs sold.

The Profit Earned per Hot Dog Sold

The owner earns a profit of $2 for each hot dog sold. This profit is a variable income that depends on the number of hot dogs sold.

The Linear Function

The profit earned by the hot dog stand can be represented by a linear function. A linear function is a function that can be written in the form:

y = mx + b

where y is the dependent variable (the profit earned), x is the independent variable (the number of hot dogs sold), m is the slope of the line (the rate of change of the profit), and b is the y-intercept (the initial profit earned).

In this case, the linear function can be written as:

y = 2x - 48

where y is the profit earned, x is the number of hot dogs sold, and 2 is the slope of the line (the rate of change of the profit).

The Equation of the Linear Function

The equation of the linear function is:

y = 2x - 48

This equation represents the profit earned by the hot dog stand as a function of the number of hot dogs sold.

Interpreting the Equation

The equation y = 2x - 48 can be interpreted as follows:

  • The profit earned by the hot dog stand is $2 for each hot dog sold.
  • The initial profit earned by the hot dog stand is $-48, which means that the owner incurs a loss of $48 each morning for the day's supply of hot dogs, buns, and mustard.
  • The profit earned by the hot dog stand increases by $2 for each hot dog sold.

Graphing the Linear Function

The linear function y = 2x - 48 can be graphed on a coordinate plane. The graph of the linear function is a straight line with a slope of 2 and a y-intercept of -48.

Solving the Equation

The equation y = 2x - 48 can be solved for x, which represents the number of hot dogs sold. To solve the equation, we can add 48 to both sides of the equation:

2x = y + 48

Dividing both sides of the equation by 2, we get:

x = (y + 48) / 2

This equation represents the number of hot dogs sold as a function of the profit earned.

Example

Suppose the owner of the hot dog stand wants to know how many hot dogs he needs to sell to earn a profit of $100. To find the number of hot dogs sold, we can substitute y = 100 into the equation x = (y + 48) / 2:

x = (100 + 48) / 2

x = 148 / 2

x = 74

Therefore, the owner needs to sell 74 hot dogs to earn a profit of $100.

Conclusion

In conclusion, the profit earned by a hot dog stand is a linear function of the number of hot dogs sold. The equation y = 2x - 48 represents the profit earned by the hot dog stand as a function of the number of hot dogs sold. The equation can be used to determine the number of hot dogs sold to earn a specific profit.

References

Glossary

  • Linear function: A function that can be written in the form y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept.
  • Slope: The rate of change of the function.
  • Y-intercept: The initial value of the function.
  • Graph: A visual representation of the function on a coordinate plane.
    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 the relationship between the number of hot dogs sold and the profit earned by a hot dog stand. We used a linear function to model this relationship and determined the equation that represents the profit earned by the hot dog stand. In this article, we will answer some frequently asked questions about the profit earned by a hot dog stand.

Q&A

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 per hot dog sold?

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

Q: What is the equation of the linear function that represents the profit earned by the hot dog stand?

A: The equation of the linear function is y = 2x - 48, where y is the profit earned, x is the number of hot dogs sold, and 2 is the slope of the line (the rate of change of the profit).

Q: How can I determine the number of hot dogs sold to earn a specific profit?

A: To determine the number of hot dogs sold to earn a specific profit, you can use the equation x = (y + 48) / 2, where x is the number of hot dogs sold and y is the profit earned.

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 loss of $48 each morning for the day's supply of hot dogs, buns, and mustard.

Q: What is the slope of the linear function?

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

Q: Can I use the linear function to determine the profit earned by the hot dog stand if I know the number of hot dogs sold?

A: Yes, you can use the linear function to determine the profit earned by the hot dog stand if you know the number of hot dogs sold. Simply substitute the number of hot dogs sold into the equation y = 2x - 48 to find the profit earned.

Q: What is the graph of the linear function?

A: The graph of the linear function is a straight line with a slope of 2 and a y-intercept of -48.

Q: Can I use the graph to determine the number of hot dogs sold to earn a specific profit?

A: Yes, you can use the graph to determine the number of hot dogs sold to earn a specific profit. Simply find the point on the graph that corresponds to the desired profit and read the number of hot dogs sold from the x-axis.

Conclusion

In conclusion, the profit earned by a hot dog stand is a linear function of the number of hot dogs sold. The equation y = 2x - 48 represents the profit earned by the hot dog stand as a function of the number of hot dogs sold. We hope that this Q&A article has helped to clarify any questions you may have had about the profit earned by a hot dog stand.

References

Glossary

  • Linear function: A function that can be written in the form y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept.
  • Slope: The rate of change of the function.
  • Y-intercept: The initial value of the function.
  • Graph: A visual representation of the function on a coordinate plane.