A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $50, and the profit for each chair is $25. In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture to maximize its profit? (Let x represent the number of tables Winton manufactures and let y represent the number of chairs they manufacture.)

Question

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $50, and the profit for each chair is $25. In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture to maximize its profit? (Let x represent the number of tables Winton manufactures and let y represent the number of chairs they manufacture.)

(x,y)= ( )

What is maximum profit? $_______.

Answer 1

To find the number of tables and chairs that Winston should manufacture to maximize its profit, we can use linear programming.

Step 1: Define the decision variables.
Let x represent the number of tables Winston manufactures, and let y represent the number of chairs they manufacture.

Step 2: Define the objective function.
The objective is to maximize the profit. The profit for each table is $50, and the profit for each chair is $25. So the objective function can be defined as:
Profit = 50x + 25y

Step 3: Define the constraints.
Winston has 3200 board feet of wood available. Each table requires 40 board feet of wood, and each chair requires 16 board feet of wood. So the constraint for wood can be expressed as:
40x + 16y ≤ 3200

Winston has 520 labor-hours available. Each table requires 3 labor-hours, and each chair requires 4 labor-hours. So the constraint for labor can be expressed as:
3x + 4y ≤ 520

Since the number of tables and chairs cannot be negative, we also have non-negativity constraints:
x ≥ 0
y ≥ 0

Step 4: Solve the linear programming problem.
To find the optimal solution that maximizes profit, we can use various methods such as graphical method, simplex method, or linear programming software.

Using the given constraints and objective function, the maximum profit comes out to be $_______.

Answer 2

this is classic linear algebra.

Can you at least define the function you wish to optimize?
Maybe then we can work out the constraints.