Dwayne Is Selling Hamburgers And Cheeseburgers. He Has 100 Burger Buns. Each Hamburger Sells For $$ 3 3 3 $, And Each Cheeseburger Sells For $$ 3.50 3.50 3.50 $. Which System Of Inequalities Represents The Number Of Hamburgers,
Introduction
Dwayne is a burger enthusiast who has decided to sell hamburgers and cheeseburgers at his local market. He has a limited supply of 100 burger buns, which he must use to make both hamburgers and cheeseburgers. Each hamburger sells for $3, and each cheeseburger sells for $3.50. In this scenario, we need to determine which system of inequalities represents the number of hamburgers and cheeseburgers that Dwayne can sell.
The Problem
Let's denote the number of hamburgers as H and the number of cheeseburgers as C. We know that Dwayne has 100 burger buns, so the total number of burgers he can make is 100. This gives us our first inequality:
H + C ≤ 100
We also know that each hamburger sells for $3 and each cheeseburger sells for $3.50. If we let R be the total revenue, then we can write the following inequality:
3H + 3.5C ≤ R
However, we need to find a system of inequalities that represents the number of hamburgers and cheeseburgers that Dwayne can sell. To do this, we need to consider the constraints on the number of burgers he can make.
Constraints on the Number of Burgers
Since Dwayne has 100 burger buns, he can make at most 100 burgers. This gives us our first constraint:
H + C ≤ 100
We also know that Dwayne cannot sell a negative number of burgers. This gives us our second constraint:
H ≥ 0 C ≥ 0
The System of Inequalities
Now that we have our constraints, we can write the system of inequalities that represents the number of hamburgers and cheeseburgers that Dwayne can sell:
H + C ≤ 100 3H + 3.5C ≤ R H ≥ 0 C ≥ 0
However, we still need to determine the value of R, which represents the total revenue. To do this, we need to consider the prices of the hamburgers and cheeseburgers.
Prices of Hamburgers and Cheeseburgers
Each hamburger sells for $3, and each cheeseburger sells for $3.50. If we let P be the price of a hamburger and Q be the price of a cheeseburger, then we can write the following equations:
P = 3 Q = 3.5
The Total Revenue
Now that we have the prices of the hamburgers and cheeseburgers, we can determine the total revenue. Let's say that Dwayne sells H hamburgers and C cheeseburgers. Then the total revenue is:
R = PH + QC R = 3H + 3.5C
The System of Inequalities with Total Revenue
Now that we have the total revenue, we can write the system of inequalities that represents the number of hamburgers and cheeseburgers that Dwayne can sell:
H + C ≤ 100 3H + 3.5C ≤ 3H + 3.5C H ≥ 0 C ≥ 0
However, the second inequality is always true, so we can simplify the system to:
H + C ≤ 100 H ≥ 0 C ≥ 0
Conclusion
In this scenario, Dwayne is selling hamburgers and cheeseburgers at his local market. He has a limited supply of 100 burger buns, which he must use to make both hamburgers and cheeseburgers. Each hamburger sells for $3, and each cheeseburger sells for $3.50. We have determined that the system of inequalities that represents the number of hamburgers and cheeseburgers that Dwayne can sell is:
H + C ≤ 100 H ≥ 0 C ≥ 0
This system of inequalities represents the constraints on the number of hamburgers and cheeseburgers that Dwayne can sell, given the limited supply of burger buns and the prices of the hamburgers and cheeseburgers.
Final Answer
The final answer is the system of inequalities:
H + C ≤ 100 H ≥ 0 C ≥ 0
This system of inequalities represents the number of hamburgers and cheeseburgers that Dwayne can sell, given the limited supply of burger buns and the prices of the hamburgers and cheeseburgers.
Introduction
In our previous article, we explored the system of inequalities that represents the number of hamburgers and cheeseburgers that Dwayne can sell. We determined that the system of inequalities is:
H + C ≤ 100 H ≥ 0 C ≥ 0
In this article, we will answer some frequently asked questions about Dwayne's burger sales and the system of inequalities.
Q: What is the meaning of the first inequality, H + C ≤ 100?
A: The first inequality, H + C ≤ 100, represents the constraint on the total number of burgers that Dwayne can make. Since he has 100 burger buns, he can make at most 100 burgers. This means that the total number of hamburgers (H) and cheeseburgers (C) that Dwayne can sell must be less than or equal to 100.
Q: What is the meaning of the second inequality, H ≥ 0?
A: The second inequality, H ≥ 0, represents the constraint on the number of hamburgers that Dwayne can sell. Since Dwayne cannot sell a negative number of hamburgers, the number of hamburgers (H) must be greater than or equal to 0.
Q: What is the meaning of the third inequality, C ≥ 0?
A: The third inequality, C ≥ 0, represents the constraint on the number of cheeseburgers that Dwayne can sell. Since Dwayne cannot sell a negative number of cheeseburgers, the number of cheeseburgers (C) must be greater than or equal to 0.
Q: How can Dwayne maximize his revenue?
A: To maximize his revenue, Dwayne should sell as many cheeseburgers as possible, since each cheeseburger sells for $3.50, which is more than the price of a hamburger. However, he must also consider the constraint on the total number of burgers that he can make, which is 100.
Q: What is the optimal solution to the system of inequalities?
A: The optimal solution to the system of inequalities is when Dwayne sells 50 hamburgers and 50 cheeseburgers. This is because it maximizes his revenue, while also satisfying the constraints on the total number of burgers that he can make.
Q: How can Dwayne use the system of inequalities to make decisions about his burger sales?
A: Dwayne can use the system of inequalities to make decisions about his burger sales by considering the constraints on the total number of burgers that he can make, as well as the prices of the hamburgers and cheeseburgers. By analyzing the system of inequalities, Dwayne can determine the optimal solution and make informed decisions about his burger sales.
Q: What are some potential applications of the system of inequalities in real-world scenarios?
A: The system of inequalities has many potential applications in real-world scenarios, such as:
- Resource allocation: The system of inequalities can be used to allocate resources, such as budget or personnel, to different projects or tasks.
- Scheduling: The system of inequalities can be used to schedule tasks or events, such as meetings or appointments.
- Optimization: The system of inequalities can be used to optimize processes or systems, such as supply chain management or inventory control.
Conclusion
In this article, we have answered some frequently asked questions about Dwayne's burger sales and the system of inequalities. We have also discussed the potential applications of the system of inequalities in real-world scenarios. By understanding the system of inequalities, Dwayne can make informed decisions about his burger sales and maximize his revenue.
Final Answer
The final answer is the system of inequalities:
H + C ≤ 100 H ≥ 0 C ≥ 0
This system of inequalities represents the number of hamburgers and cheeseburgers that Dwayne can sell, given the limited supply of burger buns and the prices of the hamburgers and cheeseburgers.