Let's assume we order x printers of type A and y printers of type B.
According to the given information, the cost of each type A printer is $237, and the profit per type A printer is $22. So, the total cost of x type A printers is 237x, and the total profit is 22x.
Similarly, the cost of each type B printer is $122, and the profit per type B printer is $19. So, the total cost of y type B printers is 122y, and the total profit is 19y.
We are given the following constraints:
1. x + y ≤ 120 (We can order no more than 120 printers)
2. 22x + 19y ≥ 2400 (We need to make at least $2400 profit)
We also have the constraint that we must order at least one of each type, so x ≥ 1 and y ≥ 1.
To minimize the cost, we need to maximize the profit. Since the profit per type A printer is higher than the profit per type B printer, it is beneficial to order more type A printers.
To find the optimal solution, we can use linear programming.
Let's solve this problem step-by-step:
Step 1: Set up the objective function:
We want to maximize the profit. So, the objective function is:
Profit = 22x + 19y
Step 2: Set up the constraints:
- x + y ≤ 120 (Constraint 1)
- 22x + 19y ≥ 2400 (Constraint 2)
- x ≥ 1 (Constraint 3)
- y ≥ 1 (Constraint 4)
Step 3: Graph the feasible region:
Let's plot the constraints on a graph:
Graph:
x-axis represents the number of type A printers (x)
y-axis represents the number of type B printers (y)
1. Constraint 1: x + y ≤ 120
To plot this inequality equation, we represent it as an equality equation: x + y = 120.
When we plot this equality equation, we get a line passing through the points (0,120) and (120,0).
But we are interested in the region below this line since x + y should be less than or equal to 120.
2. Constraint 2: 22x + 19y ≥ 2400
To plot this inequality equation, we represent it as an equality equation: 22x + 19y = 2400.
When we plot this equality equation, we get a line passing through the points (0,126.32) and (109,0).
But we are interested in the region above this line since 22x + 19y should be greater than or equal to 2400.
3. Constraint 3: x ≥ 1
This constraint represents x greater than or equal to 1.
We draw a vertical line passing through x = 1.
4. Constraint 4: y ≥ 1
This constraint represents y greater than or equal to 1.
We draw a horizontal line passing through y = 1.
The feasible region is the shaded region that satisfies all the constraints.
(Note: The graph may not be perfectly accurate due to limited visual representation.)
Step 4: Find the optimal solution:
To find the optimal solution, we need to find the point within the feasible region that maximizes the objective function (profit).
We can compare the profit values at the corner points of the feasible region:
Corner points:
A (1, 119)
B (94, 1)
C (1, 1)
Evaluate the objective function at each corner point:
Point A (1, 119):
Profit = 22(1) + 19(119) = 2303
Point B (94, 1):
Profit = 22(94) + 19(1) = 2090
Point C (1, 1):
Profit = 22(1) + 19(1) = 41
The maximum profit is achieved at point A (1, 119), where x = 1 (type A printers) and y = 119 (type B printers). Here, the profit is $2303.
Therefore, to minimize the cost and achieve at least $2400 profit, you should order 1 type A printer and 119 type B printers.