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A catering business offers two sizes of baked ziti. The small ziti dish uses 1 cup of sauce and 1.75 cups of cheese. The large ziti dish uses 2 cups of sauce and 3 cups of cheese. The business has 100 cups of sauce and 100 cups of cheese on hand. It makes [tex]$6 profit on small dishes and $[/tex]5 profit on large dishes. Let [tex]\( x \)[/tex] represent the number of small dishes and [tex]\( y \)[/tex] represent the number of large dishes.

What are the constraints for the problem?

[tex]\[
\begin{array}{l}
x + \frac{7}{4}y \leq 6 \\
2x + 3y \leq 5 \\
x \geq 0 \\
y \geq 0
\end{array}
\][/tex]

Sagot :

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.