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
Introduction
A hot dog stand is a small business that sells hot dogs to customers. The profit earned by the hot dog stand is a linear function of the number of hot dogs sold. In this article, we will discuss 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 cost, meaning that it does not change regardless of the number of hot dogs sold.
The Profit Earned per Hot Dog Sold
The owner of the hot dog stand earns a profit of $2 for each hot dog sold. This profit is a variable cost, meaning that it changes depending on the number of hot dogs sold.
The Equation of the Profit Function
The profit earned by the hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function can be represented as:
P(x) = 2x - 48
Where P(x) is the profit earned and x is the number of hot dogs sold.
Explanation of the Equation
In the equation P(x) = 2x - 48, the term 2x represents the profit earned from selling x hot dogs. The term -48 represents the fixed cost of $48 that the owner incurs each morning.
Graphing the Profit Function
The graph of the profit function is a straight line with a slope of 2 and a y-intercept of -48. The graph can be represented as:
| x | P(x) |
|---|---|
| 0 | -48 |
| 10 | 12 |
| 20 | 32 |
| 30 | 52 |
| 40 | 72 |
| 50 | 92 |
Interpretation of the Graph
The graph of the profit function shows that the profit earned by the hot dog stand increases by $2 for each hot dog sold. The graph also shows that the profit earned by the hot dog stand is negative when the number of hot dogs sold is 0, indicating that the owner incurs a loss when no hot dogs are sold.
Conclusion
In conclusion, the profit earned by a hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function is P(x) = 2x - 48, where P(x) is the profit earned and x is the number of hot dogs sold. The graph of the profit function is a straight line with a slope of 2 and a y-intercept of -48.
Real-World Applications
The concept of a linear function can be applied to many real-world situations, such as:
- Business: The profit earned by a business is a linear function of the number of products sold.
- Economics: The demand for a product is a linear function of the price of the product.
- Science: The distance traveled by an object is a linear function of the time traveled.
Solved Problems
Problem 1
A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells 20 hot dogs, what is the profit earned?
Solution
The profit earned by the hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function is P(x) = 2x - 48, where P(x) is the profit earned and x is the number of hot dogs sold.
P(20) = 2(20) - 48 = 40 - 48 = -8
The profit earned by the hot dog stand is -$8 when 20 hot dogs are sold.
Problem 2
A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells 30 hot dogs, what is the profit earned?
Solution
The profit earned by the hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function is P(x) = 2x - 48, where P(x) is the profit earned and x is the number of hot dogs sold.
P(30) = 2(30) - 48 = 60 - 48 = 12
The profit earned by the hot dog stand is $12 when 30 hot dogs are sold.
Practice Problems
Problem 1
A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells x hot dogs, what is the profit earned?
Solution
The profit earned by the hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function is P(x) = 2x - 48, where P(x) is the profit earned and x is the number of hot dogs sold.
P(x) = 2x - 48
Problem 2
A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells 40 hot dogs, what is the profit earned?
Solution
The profit earned by the hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function is P(x) = 2x - 48, where P(x) is the profit earned and x is the number of hot dogs sold.
P(40) = 2(40) - 48 = 80 - 48 = 32
The profit earned by the hot dog stand is $32 when 40 hot dogs are sold.
Conclusion
Q: What is the profit earned by a hot dog stand?
A: The profit earned by a hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function is P(x) = 2x - 48, where P(x) is the profit earned and x is the number of hot dogs sold.
Q: What is the fixed cost of running a hot dog stand?
A: The fixed cost of running a 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 slope of the profit function?
A: The slope of the profit function is 2, which means that the profit earned increases by $2 for each hot dog sold.
Q: What is the y-intercept of the profit function?
A: The y-intercept of the profit function is -48, which means that the profit earned is negative when no hot dogs are sold.
Q: How can the profit function be used in real-world applications?
A: The profit function can be used in many real-world applications, such as:
- Business: The profit earned by a business is a linear function of the number of products sold.
- Economics: The demand for a product is a linear function of the price of the product.
- Science: The distance traveled by an object is a linear function of the time traveled.
Q: How can the profit function be graphed?
A: The profit function can be graphed as a straight line with a slope of 2 and a y-intercept of -48.
Q: What are some solved problems related to the profit function?
A: Some solved problems related to the profit function include:
- Problem 1: A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells 20 hot dogs, what is the profit earned?
- Problem 2: A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells 30 hot dogs, what is the profit earned?
Q: What are some practice problems related to the profit function?
A: Some practice problems related to the profit function include:
- Problem 1: A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells x hot dogs, what is the profit earned?
- Problem 2: A hot dog stand sells hot dogs for $2 each. The owner incurs a cost of $48 each morning for the day's supply of hot dogs, buns, and mustard. If the owner sells 40 hot dogs, what is the profit earned?
Q: What is the conclusion of the article?
A: In conclusion, the profit earned by a hot dog stand is a linear function of the number of hot dogs sold. The equation of the profit function is P(x) = 2x - 48, where P(x) is the profit earned and x is the number of hot dogs sold. The graph of the profit function is a straight line with a slope of 2 and a y-intercept of -48.