A bicycle builder makes two models. The basic model requires 2 hours of frame construction, 4 hours of assembly, and 1 hour of finishing. The deluxe model requires 3 hours of frame construction, 3 hours of assembly, and 2 hours of finishing. Each day 36 hours are available for frame construction, 40 hours for assembly, and 20 hours for finishing. If the profit on the basic model is $100 and on the deluxe model is $150, how many of each should be made to maximize the profit?
To solve this problem, we need to determine the number of basic and deluxe models that should be made in order to maximize profit. Let's assume that the number of basic models made is "x" and the number of deluxe models made is "y".
The profit from the basic model is $100, and the profit from the deluxe model is $150. Therefore, the total profit can be calculated using the following equation:
Profit = (Profit per Basic Model) * (Number of Basic Models) + (Profit per Deluxe Model) * (Number of Deluxe Models)
Profit = 100x + 150y
Now, let's set up the constraints based on the available hours for each task:
1. Frame Construction: The basic model takes 2 hours and the deluxe model takes 3 hours. The total available hours for frame construction are 36 hours.
2 hours * x (number of basic models) + 3 hours * y (number of deluxe models) ≤ 36
2x + 3y ≤ 36
2. Assembly: The basic model requires 4 hours, and the deluxe model requires 3 hours. The total available hours for assembly are 40 hours.
4 hours * x + 3 hours * y ≤ 40
4x + 3y ≤ 40
3. Finishing: The basic model takes 1 hour, and the deluxe model takes 2 hours. The total available hours for finishing are 20 hours.
1 hour * x + 2 hours * y ≤ 20
x + 2y ≤ 20
Now, we have a system of linear inequalities:
2x + 3y ≤ 36
4x + 3y ≤ 40
x + 2y ≤ 20
We can solve this system using graphing, substitution, or elimination methods. In this explanation, let's solve it using the graphing method:
1. Graph each inequality on a scatter plot.
2. Find the region where all the inequalities overlap. This region represents the feasible solutions for the problem.
3. Identify the corner points of the feasible region, as these are the potential solutions to maximize profit.
4. Substitute each corner point into the profit equation and find the maximum profit.
After solving the system of inequalities using the graphing method, we find that the feasible region has four corner points: (0, 12), (6, 4), (8, 0), and (0, 0).
To find the maximum profit, substitute each of these corner points into the profit equation:
Profit = 100x + 150y
For the corner points: (0, 12) → Profit = 100(0) + 150(12) = $1,800
(6, 4) → Profit = 100(6) + 150(4) = $1,200
(8, 0) → Profit = 100(8) + 150(0) = $800
(0, 0) → Profit = 100(0) + 150(0) = $0
From these calculations, we can see that the maximum profit of $1,800 is achieved when 0 basic models and 12 deluxe models are made.
Therefore, to maximize profit, the bicycle builder should make 0 basic models and 12 deluxe models.