To determine the constraints for this problem involving a catering business that offers two sizes of baked ziti, let's carefully analyze the problem step-by-step and identify each relevant constraint.
1. Sauce Constraint:
Each small ziti dish uses 1 cup of sauce.
Each large ziti dish uses [tex]\(1 \frac{3}{4}\)[/tex] cups of sauce. Converting the mixed number to an improper fraction, we get [tex]\(1 \frac{3}{4} = 1.75\)[/tex] cups of sauce.
Therefore, if [tex]\(x\)[/tex] is the number of small dishes and [tex]\(y\)[/tex] is the number of large dishes, the total amount of sauce used is:
[tex]\[
x + 1.75y
\][/tex]
Since the business has 10 cups of sauce on hand, the constraint for the sauce can be expressed as:
[tex]\[
x + 1.75y \leq 10
\][/tex]
2. Cheese Constraint:
Each small ziti dish uses 2 cups of cheese.
Each large ziti dish uses 3 cups of cheese.
Therefore, if [tex]\(x\)[/tex] is the number of small dishes and [tex]\(y\)[/tex] is the number of large dishes, the total amount of cheese used is:
[tex]\[
2x + 3y
\][/tex]
Since the business has 10 cups of cheese on hand, the constraint for the cheese can be expressed as:
[tex]\[
2x + 3y \leq 10
\][/tex]
3. Non-negativity Constraints:
The number of small dishes ([tex]\(x\)[/tex]) and the number of large dishes ([tex]\(y\)[/tex]) must be non-negative, since negative quantities do not make sense in this context. Therefore, we have two more constraints:
[tex]\[
x \geq 0
\][/tex]
[tex]\[
y \geq 0
\][/tex]
Summarizing the constraints, we have:
[tex]\[
\begin{cases}
x + 1.75y \leq 10 \\
2x + 3y \leq 10 \\
x \geq 0 \\
y \geq 0
\end{cases}
\][/tex]
These represent the feasible region within which the business must operate to maximize their profit while adhering to the available resources.
