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Answer :
To determine the constraints for this problem, we need to consider the available resources and how they are used in each special:
1. Bread Usage Constraint:
Each sandwich, regardless of whether it's roast beef or turkey, uses two slices of bread. We have a total of 120 slices available. So, for the roast beef specials (represented by [tex]\( x \)[/tex]) and the turkey specials (represented by [tex]\( y \)[/tex]), the constraint for bread is:
[tex]\[
2x + 2y \leq 120
\][/tex]
2. Cheese Usage Constraint:
The roast beef sandwich uses three slices of cheese, and the turkey special uses four slices of cheese. With 160 slices of cheese available, the cheese constraint is:
[tex]\[
3x + 4y \leq 160
\][/tex]
3. Non-negativity Constraints:
We cannot make a negative number of sandwiches, so both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] must be non-negative:
[tex]\[
x \geq 0
\][/tex]
[tex]\[
y \geq 0
\][/tex]
Putting all these constraints together, we have:
- Bread Constraint: [tex]\( 2x + 2y \leq 120 \)[/tex]
- Cheese Constraint: [tex]\( 3x + 4y \leq 160 \)[/tex]
- Non-negativity for Roast Beef Specials: [tex]\( x \geq 0 \)[/tex]
- Non-negativity for Turkey Specials: [tex]\( y \geq 0 \)[/tex]
With these constraints, the deli can maximize its profits while ensuring it doesn't exceed its supply of bread and cheese.
1. Bread Usage Constraint:
Each sandwich, regardless of whether it's roast beef or turkey, uses two slices of bread. We have a total of 120 slices available. So, for the roast beef specials (represented by [tex]\( x \)[/tex]) and the turkey specials (represented by [tex]\( y \)[/tex]), the constraint for bread is:
[tex]\[
2x + 2y \leq 120
\][/tex]
2. Cheese Usage Constraint:
The roast beef sandwich uses three slices of cheese, and the turkey special uses four slices of cheese. With 160 slices of cheese available, the cheese constraint is:
[tex]\[
3x + 4y \leq 160
\][/tex]
3. Non-negativity Constraints:
We cannot make a negative number of sandwiches, so both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] must be non-negative:
[tex]\[
x \geq 0
\][/tex]
[tex]\[
y \geq 0
\][/tex]
Putting all these constraints together, we have:
- Bread Constraint: [tex]\( 2x + 2y \leq 120 \)[/tex]
- Cheese Constraint: [tex]\( 3x + 4y \leq 160 \)[/tex]
- Non-negativity for Roast Beef Specials: [tex]\( x \geq 0 \)[/tex]
- Non-negativity for Turkey Specials: [tex]\( y \geq 0 \)[/tex]
With these constraints, the deli can maximize its profits while ensuring it doesn't exceed its supply of bread and cheese.
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