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.
To determine the optimal combination of dining chairs and rocking chairs that maximizes profit, we can use the concept of linear programming.
Let's represent the number of dining chairs as x and the number of rocking chairs as y.
Our objective is to maximize profit, which is given by the equation:
Profit = 60x + 90y
However, we have two constraints:
1) Total chairs constraint: x + y = 30
2) Total available work hours constraint: 1x + 2y ≤ 40
Now, let's plot these constraints on a graph:
The total chairs constraint can be represented by a straight line passing through the points (0,30) and (30,0).
The total available work hours constraint can be represented by a straight line passing through the points (0,20) and (40,0).
We need to find the feasible region which satisfies both constraints.
After graphing the constraints, we see that the feasible region is the triangle formed by points (0, 20), (20, 10), and (25, 5).
Next, we need to evaluate the objective function at the vertices of the feasible region to determine the combination of chairs that results in maximum profit.
At vertex (0, 20): Profit = 60(0) + 90(20) = $1800
At vertex (20, 10): Profit = 60(20) + 90(10) = $2400
At vertex (25, 5): Profit = 60(25) + 90(5) = $2700
From the calculations, we find that the maximum profit can be achieved by making 25 dining chairs and 5 rocking chairs, for a total profit of $2700.
Therefore, the correct statement is:
A. The carpenter can maximize profits by making 25 dining chairs and 5 rocking chairs.