• He has enough wood to make 30 chairs.
• He makes $60 profit on a dining chair and $90 profit on a rocking chair.
• It takes him 1 hour to make a dining chair and 2 hours to make a rocking chair.
• He only has 40 hours available to work on the chairs.
The carpenter wants to maximize his profit given the constraints. He draws the graph below to represent this situation.
Use the drop-down menus to correctly complete the statement below.
Given the constraints, the carpenter can maximize profits by making blank dining chairs and blank rocking chairs
Given the constraints, the carpenter can maximize profits by making 20 dining chairs and 10 rocking chairs.
Given the constraints, the carpenter can maximize profits by making 25 dining chairs and 5 rocking chairs.
To find the number of dining chairs and rocking chairs that the carpenter should make in order to maximize his profits, we can use linear programming. Linear programming involves setting up a system of equations and inequalities to represent the constraints and the objective function.
Let's denote the number of dining chairs as 'x' and the number of rocking chairs as 'y'. We want to find the values of 'x' and 'y' that maximize the profit.
Based on the given information, we have the following constraints:
1. The carpenter has enough wood to make 30 chairs, so the total number of chairs should be:
x + y ≤ 30
2. The carpenter has 40 hours available to work. It takes 1 hour to make a dining chair and 2 hours to make a rocking chair. Therefore, the total hours worked should be:
x + 2y ≤ 40
To maximize the profit, we need to find the objective function. The profit from each dining chair is $60, and the profit from each rocking chair is $90. So, the objective function can be defined as:
Profit = 60x + 90y
Now, to find the number of dining chairs and rocking chairs that maximize the profit, we need to solve this system of inequalities using the graphical method.