Alright, let's fill in the details based on the information given:
### Constraints
1. Total time constraint for Machine A:
[tex]\[
2 \cdot x + 3 \cdot y \leq 36
\][/tex]
2. Total time constraint for Machine B:
[tex]\[
4 \cdot y \leq 42
\][/tex]
3. Total time constraint for Machine C:
[tex]\[
2 \cdot x + y \leq 20
\][/tex]
4. Non-negativity constraints for the number of hats and scarves:
[tex]\[
x \geq 0
\][/tex]
[tex]\[
y \geq 0
\][/tex]
### Vertices of the Feasible Region
- (0, 10.5)
- (3, 8)
- (9, 0)
- (0, 0)
### Optimization Equation
The profit function is:
[tex]\[
\text{Profit} = 5 \cdot x + 4 \cdot y
\][/tex]
### Maximum Profit
The given vertices and corresponding profits are:
- At (0, 10.5): [tex]\(\text{Profit} = 5 \cdot 0 + 4 \cdot 10.5 = 42.0\)[/tex]
- At (3, 8): [tex]\(\text{Profit} = 5 \cdot 3 + 4 \cdot 8 = 47\)[/tex]
- At (9, 0): [tex]\(\text{Profit} = 5 \cdot 9 + 4 \cdot 0 = 45\)[/tex]
- At (0, 0): [tex]\(\text{Profit} = 5 \cdot 0 + 4 \cdot 0 = 0\)[/tex]
The maximum profit of [tex]\(\$47\)[/tex] occurs at the point (3, 8).
### Conclusion
Alice's maximum profit is [tex]\(\$47\)[/tex] per week. She should make 3 hats and 8 scarves each week.
