scott owns a manufacturing company that produces two models of entertainment centers. the athens requires 4 feet of fancy molding and takes 4 hours to manufacture. The barcelona needs 15 feet of molding and 3 hours to manufacture. In a given week, there are 120 hours of labor available and the company has 360 feet of molding to use for the entertainment centers. The company makes a profit of $9 on the athens and $12 on the barcelona. How many of each model should the company manufacture to maximize its profit?

1. Graph the solution of linear inequalities , indicate the feasible region, and identify the corners/vertices o the feasible region.
2. Solve the problem

Someone please help with this problem

To solve this problem, we can use linear programming. Linear programming involves maximizing or minimizing a linear objective function while satisfying a set of linear constraints.

Step 1: Define the decision variables
Let's define our decision variables:
- Let A be the number of Athens models produced
- Let B be the number of Barcelona models produced

Step 2: Formulate the objective function
We want to maximize the profit, which is given by:
Profit = 9A + 12B

Step 3: Formulate the constraints
We have the following constraints:
- Labor constraint: Each Athens model takes 4 hours, and each Barcelona model takes 3 hours. The total available labor is 120 hours, so the labor constraint can be written as:
4A + 3B ≤ 120

- Molding constraint: Each Athens model requires 4 feet of molding, and each Barcelona model requires 15 feet. The total available molding is 360 feet, so the molding constraint can be written as:
4A + 15B ≤ 360

- Non-negativity constraint: We cannot produce a negative number of either model, so A ≥ 0 and B ≥ 0.

Step 4: Graph the solution of linear inequalities
To graph the solution, we can plot the feasible region based on the constraints. Start by graphing the equations for the constraints as if they were equalities. Then shade the region that satisfies the inequalities (i.e., the region below the lines).

Step 5: Identify the corners/vertices of the feasible region
The corners or vertices of the feasible region are the points where the lines intersect. These points represent different combinations of Athens and Barcelona models that can be produced.

Step 6: Solve the problem
To find the optimal solution, substitute the values of the corners/vertices into the objective function and calculate the profit. The corner/vertex with the highest profit will give us the optimal number of Athens and Barcelona models to produce.

Using this approach, you should be able to determine the number of each model that the company should manufacture to maximize its profit.