A carpenter makes wooden chairs. 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 has only 40 hours available to work on the chairs. The carpenter wants to maximize his profit given the constraints listed above. He draws the following graph to represent this situation. Given the constraints, which statement is true? A. The carpenter can maximize profits by making 25 dining chairs and 5 rocking chairs. B. The carpenter can maximize profits by making 0 dining chairs and 20 rocking chairs. C. The carpenter can maximize profits by making 20 dining chairs and 10 rocking chairs. D. The carpenter can maximize profits by making 30 dining chairs and 0 rocking chairs

Question

A carpenter makes wooden chairs. 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 has only 40 hours available to work on the chairs. The carpenter wants to maximize his profit given the constraints listed above. He draws the following graph to represent this situation. Given the constraints, which statement is true? A. The carpenter can maximize profits by making 25 dining chairs and 5 rocking chairs. B. The carpenter can maximize profits by making 0 dining chairs and 20 rocking chairs. C. The carpenter can maximize profits by making 20 dining chairs and 10 rocking chairs. D. The carpenter can maximize profits by making 30 dining chairs and 0 rocking chairs

Answer 1

To answer this question, we can use the concept of linear programming. The objective is to maximize profits, and the constraints are the availability of wood and the carpenter's working hours.

Let's assign variables:
- Let x be the number of dining chairs.
- Let y be the number of rocking chairs.

From the information given:
1. The carpenter has enough wood to make 30 chairs. Therefore, the constraint for the number of chairs is: x + y ≤ 30.
2. The carpenter makes $60 profit on each dining chair and $90 profit on each rocking chair. Therefore, the objective function for profit is: Profit = 60x + 90y.
3. It takes 1 hour to make a dining chair and 2 hours to make a rocking chair. Therefore, the constraint for working hours is: x + 2y ≤ 40.

Now let's analyze the answer choices:

A. The carpenter can maximize profits by making 25 dining chairs and 5 rocking chairs.
Profit: 60(25) + 90(5) = $1950
Working hours: 25 + 2(5) = 35
This choice satisfies all the constraints and gives a certain profit.

B. The carpenter can maximize profits by making 0 dining chairs and 20 rocking chairs.
Profit: 60(0) + 90(20) = $1800
Working hours: 0 + 2(20) = 40
This choice satisfies all the constraints and gives a certain profit.

C. The carpenter can maximize profits by making 20 dining chairs and 10 rocking chairs.
Profit: 60(20) + 90(10) = $2400
Working hours: 20 + 2(10) = 40
This choice satisfies all the constraints and gives a certain profit.

D. The carpenter can maximize profits by making 30 dining chairs and 0 rocking chairs.
Profit: 60(30) + 90(0) = $1800
Working hours: 30 + 2(0) = 30
This choice does not satisfy the constraint for working hours.

Therefore, the statement that is true is: C. The carpenter can maximize profits by making 20 dining chairs and 10 rocking chairs.