Your computer-supply store sells two types of inkjet printers. The first, type A, costs $237 and you make a $22 profit on each one. The second, type B, costs $122 and you make a $19 profit on each one. You can order no more than 120 printers this month, and you need to make at least $2,400 profit on them. If you must order at least one of each type of printer, how many of each type of printer should you order if you want to minimize your cost?
A. 69 of type A : 51 of type B
B. 40 of type A : 80 of type B
C. 51 of type A : 69 of type B
D. 80 of type A : 40 of type B
To solve the problem, let's assume that 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 on each one is $22.
- The cost of each type B printer is $122 and the profit on each one is $19.
- The maximum number of printers that can be ordered is 120.
- The minimum required profit from the printers is $2,400.
To minimize cost and meet the requirements, we need to set up the following equations:
Cost equation: (x * 237) + (y * 122) ≤ Total budget
Profit equation: (x * 22) + (y * 19) ≥ 2,400
We also have the following constraints:
- x + y ≤ 120 (Maximum number of printers)
- x ≥ 1 (At least one of each type)
To solve this problem, we can use linear programming techniques. However, since only integer values are allowed for the number of printers, we will use brute-force to find the optimal solution.
Trying out different combinations, we find that option C. 51 of type A and 69 of type B satisfies all the requirements while minimizing the cost.
Therefore, the answer is C. 51 of type A : 69 of type B.
Let's first assign variables to the number of printers of each type we will order. Let x denote the number of type A printers and y denote the number of type B printers.
Since we must order at least one of each type, we know that x > 0 and y > 0.
Now let's set up the constraints based on the given information:
1) x + y ≤ 120 (We can order no more than 120 printers.)
2) 22x + 19y ≥ 2400 (We need to make at least $2,400 profit on the printers.)
To minimize the cost, we need to minimize the total cost function C(x, y), which is given by:
C(x, y) = 237x + 122y
To solve this linear program, we can use a graphical method or a mathematical method called linear programming. Here, we will use the graphical method.
First, we will plot the feasible region on a graph based on the constraints.
For constraint 1: x + y ≤ 120
We can rewrite this constraint as y ≤ -x + 120 and plot the corresponding line on the graph.
Now, we will plot the other constraint on the same graph.
For constraint 2: 22x + 19y ≥ 2400
We can rewrite this constraint as y ≥ (-22/19)x + (2400/19) and plot the corresponding line on the graph.
The graph will show the feasible region, and the point at which the total cost function C(x, y) has the minimum value will give us the optimal solution.
After plotting the feasible region, we see that the coordinates that minimize the cost fall between the points (51, 69) and (69, 51).
Therefore, the answer is option A: 69 of type A and 51 of type B.
To solve this problem, we can set up a system of equations.
Let x be the number of type A printers and y be the number of type B printers.
We know that the total number of printers ordered can be represented by the equation:
x + y ≤ 120
We also know that the total profit made on the printers must be at least $2,400, so we have:
22x + 19y ≥ 2,400
Since we must order at least one of each type of printer, the domain for x and y is:
x ≥ 1
y ≥ 1
To minimize the cost, we want to minimize the following expression:
Total cost = 237x + 122y
Now, we can solve this system of inequalities and find the solution that minimizes the cost.
Step 1: Graph the system of inequalities on a coordinate plane.
Plot the line x + y = 120. This line represents the constraint that the total number of printers ordered cannot exceed 120.
Plot the line 22x + 19y = 2,400. This line represents the constraint that the total profit must be at least $2,400.
Shade the area below or on the line x + y = 120 to represent x + y ≤ 120.
Shade the area above or on the line 22x + 19y = 2,400 to represent 22x + 19y ≥ 2,400.
Step 2: Find the feasible region.
The feasible region is the overlapping shaded region between the two lines.
Step 3: Find the corner points of the feasible region.
The corner points of the feasible region are the vertices of the overlapping shaded region.
Step 4: Evaluate the objective function at each corner point.
Plug the x and y values of each corner point into the objective function, Total cost = 237x + 122y, and evaluate the expression to find the total cost at each corner point.
Step 5: Identify the corner point with the minimum cost.
Compare the total costs at each corner point and identify the corner point with the minimum cost.
Based on the calculations, the corner point with the minimum cost is Option A. 69 type A printers and 51 type B printers. Therefore, the answer is:
A. 69 of type A : 51 of type B.