To maximize revenue, we need to determine the number of rocking chairs and porch swings that should be made per week. Let's solve this problem step-by-step.
Let's start by assigning variables:
Let x = the number of rocking chairs made per week
Let y = the number of porch swings made per week
Next, we need to set up some equations based on the given information:
1. Sam's working hours constraint:
Since each rocker requires 3 hours of work from Sam and each swing requires 2 hours of work, we can write the equation:
3x + 2y ≤ 40
2. Revenue equation:
The revenue from the rocking chairs can be calculated by multiplying the selling price ($160) by the number of rocking chairs (x). Similarly, the revenue from the porch swings can be calculated by multiplying the selling price ($100) by the number of porch swings (y). The total revenue is the sum of these two:
Revenue = 160x + 100y
Now, we need to find the maximum value of the revenue. To do this, we can use a technique called linear programming.
We need to graph the feasible region represented by the inequality 3x + 2y ≤ 40. The feasible region is the area on or below the line 3x + 2y = 40. This line passes through the points (0,20) and (13.3,0).
Next, we evaluate the revenue equation at each corner point of the feasible region. The corner points are where the feasible region intersects the axes or each other.
By substituting the corner points into the revenue equation, we can determine which combination of x and y will maximize the revenue.
Let's calculate the revenue at each corner point:
1. (0, 20):
Revenue = 160(0) + 100(20) = 2000
2. (13.3, 0):
Revenue = 160(13.3) + 100(0) = 2128
3. (0, 0):
Revenue = 160(0) + 100(0) = 0
Since the revenue at (13.3, 0) is the highest, the maximum revenue will be achieved by making approximately 13 rocking chairs and no porch swings per week.
Therefore, to maximize revenue, approximately 13 rocking chairs should be made per week and no porch swings should be made.