Answer:
[tex]\textsf{a)} \quad \begin{cases}s+e \leq 8\\8s+2e \geq 40\end{cases}[/tex]
[tex]\textsf{b)} \quad y = 30s + 35e[/tex]
c) 4 simple vases and 4 elaborate vases maximize profit.
The maximum profit is $260.
Step-by-step explanation:
Given information:
- A glass blower can form 8 simple vases or 2 elaborate vases in an hour.
- In a work shift of no more than 8 hours, the worker must form at least 40 vases.
Part (a)
Define the variables:
- Let s = the number of hours forming simple vases.
- Let e = the number of hours forming elaborate vases.
Create a system of inequalities using the given information and defined variables:
[tex]\begin{cases}s+e \leq 8\\8s+2e \geq 40\end{cases}[/tex]
Part (b)
Given information:
- $30 = profit per hour for the simple vases.
- $35 = profit per hour for the elaborate vases.
Let y be the total profit in dollars:
[tex]y = 30s + 35e[/tex]
Part (c)
[tex]\begin{cases}s+e \leq 8\\8s+2e \geq 40\end{cases}[/tex]
To find the number of hours the worker should spend on each type of vase to maximize profit, find the point of intersection of the two equations.
Isolate e in the first equation:
[tex]\implies e\leq8-s[/tex]
Isolate e in the second equation:
[tex]\implies 2e \geq 40-8s[/tex]
[tex]\implies e \geq 20-4s[/tex]
Equate the two expressions for e and solve for s:
[tex]\implies 8-s=20-4s[/tex]
[tex]\implies 3s=12[/tex]
[tex]\implies s=4[/tex]
Therefore, the number of hours the worker should spend on each type of vase to maximize profit is:
- Simple vases = 4 hours.
- Elaborate vases = 4 hours.
Substitute the values of s and e into the function from part (b):
[tex]\implies y=30(4)+35(4)[/tex]
[tex]\implies y=120+140[/tex]
[tex]\implies y=260[/tex]
Therefore, the maximum profit is $260.
