Linear Programming
Your club is making boxes and plant hangers to sell as a fundraiser. Your profit is $10 for a box and $8 for a hanger. How many boxes and hangers should you make in order to maximize your profit under the following conditions
A) lumber yard doantes enough wood for 30 boxes
B) craft store donates rope for 40 hangers
C)takes 4 hours to make a box and 3 hours to make a hanger
D) 9 members will work 20 hours each to make the boxes and hangers
To solve this problem using linear programming, we need to formulate an objective function and constraints.
Objective Function:
Let's define our objective function "P" as the profit we want to maximize. The profit from selling a box is $10, and the profit from selling a hanger is $8. Therefore, our objective function can be written as:
P = 10x + 8y
Where "x" represents the number of boxes produced, and "y" represents the number of hangers produced.
Constraints:
We have several constraints based on the given conditions:
A) Lumber yard donates enough wood for 30 boxes:
The number of boxes produced, x, must be less than or equal to 30.
B) Craft store donates rope for 40 hangers:
The number of hangers produced, y, must be less than or equal to 40.
C) It takes 4 hours to make a box and 3 hours to make a hanger:
The total number of hours worked, H, is given by:
H = 4x + 3y
The total number of hours worked, H, cannot exceed the available hours, which is 9 members working 20 hours each:
H ≤ 9 * 20
Now that we have the objective function and constraints, we can solve this linear programming problem using a variety of methods such as the graphical method, simplex method, or other optimization techniques.