Dwayne is selling hamburgers and cheeseburgers. He has 100 burger buns. Each hamburger sells for $3, and each cheeseburger sells for $3.50. Which system of inequalities represents the number of hamburgers, $h$, and the number of cheeseburgers, $c$, he must sell to have sales of at least $80?
$\begin{aligned} h+c & \leq 80 \\ 3 h+3.5 c & \leq 100\end{aligned}$
$h+c \leq 80$
$3 h+3.5 c \geq 100$
$h+c \leq 100$
$3 h+3.5 c \leq 80$
$h+c \leq 100$
$3 h+3.5 c \geq 80$
Answer (1)
The number of hamburgers and cheeseburgers must be less than or equal to the number of buns: h + c ≤ 100 .
The total sales from hamburgers and cheeseburgers must be at least $80 : 3 h + 3.5 c ≥ 80 .
Combine the two inequalities to form the system of inequalities.
The correct system of inequalities is: h + c ≤ 100 and 3 h + 3.5 c ≥ 80 .
Explanation
Problem Analysis Let's analyze the problem. Dwayne is selling hamburgers and cheeseburgers. We need to determine a system of inequalities that represents the constraints on the number of hamburgers ( h ) and cheeseburgers ( c ) he needs to sell to meet certain conditions.
Bun Constraint First, consider the constraint on the number of burger buns. He has 100 burger buns. Each hamburger and cheeseburger requires one bun. Therefore, the total number of hamburgers and cheeseburgers must be less than or equal to 100. This can be represented by the inequality: h + c ≤ 100
Sales Constraint Next, consider the constraint on the sales amount. Each hamburger sells for $3 and each cheeseburger sells for $3.50 . He wants to have sales of at least $80 . This can be represented by the inequality: 3 h + 3.5 c ≥ 80
System of Inequalities Combining these two inequalities, we get the following system of inequalities: { h + c ≤ 100 3 h + 3.5 c ≥ 80
Final Answer Comparing this system of inequalities with the given options, we find that the correct option is:
h + c ≤ 100 3 h + 3.5 c ≥ 80
Examples
Imagine you're running a lemonade stand. You have a limited amount of lemons and sugar, and you want to make at least a certain amount of money. Setting up a system of inequalities helps you figure out how many cups of regular lemonade and premium lemonade you need to sell, given your limited resources and desired profit. This is similar to Dwayne figuring out how many hamburgers and cheeseburgers he needs to sell with his limited buns and target sales amount. Understanding these constraints ensures you make the most efficient use of your resources to achieve your goals.
