Answer:
To attain the maximum profit, assemble 9 chain saws and 1 wood chipper.
Explanation:
To find the number of chain saws and wood chippers that should be assembled for maximum profit, we can set up a linear programming problem.
Let's assume x represents the number of chain saws to be assembled, and y represents the number of wood chippers to be assembled.
We have the following constraints:
1) Assembly time constraint: 2x + 6y ≤ 24 (maximum of 24 hours of assembly time available)
2) Non-negativity constraint: x ≥ 0, y ≥ 0 (we cannot have negative quantities)
We want to maximize the profit, which is given by the objective function:
Profit = 150x + 240y
To solve this linear programming problem, we can use graphical or algebraic methods. Here, we will use algebraic methods.
First, let's graph the feasible region determined by the constraints:
The assembly time constraint can be rewritten as:
2x + 6y ≤ 24
x + 3y ≤ 12
y ≤ (12 - x)/3
Next, we plot the boundary lines and shade the feasible region:
Feasible region:
x ≥ 0 (non-negativity constraint)
y ≥ 0 (non-negativity constraint)
y ≤ (12 - x)/3 (assembly time constraint)
Based on the graph, we can see that the feasible region is a triangular region bounded by the x-axis, y-axis, and the line y = (12 - x)/3.
To find the vertices of this feasible region, we can solve the equations:
x = 0
y = 0
y = (12 - x)/3
Solving these equations, we find the vertices:
(0, 0)
(0, 4)
(6, 0)
(9, 1)
Now, we evaluate the objective function (Profit = 150x + 240y) at each vertex to determine the maximum profit:
At (0, 0): Profit = 150(0) + 240(0) = 0
At (0, 4): Profit = 150(0) + 240(4) = 960
At (6, 0): Profit = 150(6) + 240(0) = 900
At (9, 1): Profit = 150(9) + 240(1) = 1,710
The maximum profit is attained at (9, 1), which means assembling 9 chain saws and 1 wood chipper will result in the maximum profit.
To attain the maximum profit, assemble 9 chain saws and 1 wood chipper.