Question 1172947
Absolutely! Let's break down this problem step-by-step.

**1. Define Variables**

* Let `x` be the number of units produced on machine X.
* Let `y` be the number of units produced on machine Y.

**2. Formulate Constraints**

* **Production Requirement:** x + y = 40 (The company needs to produce 40 units).
* **Machine X Time Constraint:** 30x ≤ 600 (Machine X has 10 hours = 600 minutes available). This simplifies to x ≤ 20.
* **Non-negativity:** x ≥ 0, y ≥ 0 (You can't produce a negative number of units).

**3. Formulate the Objective Function (Cost Function)**

* Cost = (30 minutes/unit * $2/minute) * x + (25 minutes/unit * $3/minute) * y
* Cost = 60x + 75y

**4. Graph the Constraints**

* **x + y = 40:**
    * When x = 0, y = 40.
    * When y = 0, x = 40.
* **x = 20:** This is a vertical line at x = 20.
* **x ≥ 0, y ≥ 0:** This limits the solution to the first quadrant.

**5. Find the Feasible Region**

* The feasible region is the area on the graph that satisfies all the constraints. It will be a polygon bounded by the constraint lines.

**6. Draw the Isocost Lines**

* We'll draw lines representing different cost levels:
    * $2,500: 60x + 75y = 2500
    * $2,700: 60x + 75y = 2700
    * $3,000: 60x + 75y = 3000
    * $3,500: 60x + 75y = 3500

To graph them, find the x and y intercepts of each line.

* 2500:
    * x=0, y = 2500/75 = 33.33
    * y=0, x = 2500/60 = 41.67
* 2700:
    * x=0, y = 2700/75 = 36
    * y=0, x = 2700/60 = 45
* 3000:
    * x=0, y = 3000/75 = 40
    * y=0, x = 3000/60 = 50
* 3500:
    * x=0, y = 3500/75 = 46.67
    * y=0, x = 3500/60 = 58.33

**7. Find the Optimal Solution**

* The optimal solution is the point within the feasible region where the isocost line is lowest (minimizing cost).
* The corner points of the feasible region are critical.
* Corner points of the feasible region:
    * (20,0)
    * (20,20)
    * (0,40)

* Test the corner points in the cost function:
    * (20,0): 60(20) + 75(0) = $1200
    * (20,20): 60(20) + 75(20) = $2700
    * (0,40): 60(0) + 75(40) = $3000

* The minimum cost is $1200, which occurs at (20,0).

**8. Analyze Isocost Lines**

* **Feasible and Suboptimal:**
    * $2,700: This line intersects the feasible region, but it's not the lowest cost line.
    * $3,000: This line also intersects the feasible region, but is not the lowest cost.
* **Feasible and Optimal:**
    * The isocost line that goes through the point (20,0) will be the optimal. The cost of 1200 is too low to be one of the isocost lines provided. We can create the 1200 isocost line by calculating the intercept points.
        * x=0, y = 1200/75 = 16
        * y=0, x = 1200/60 = 20
* **Not Feasible:**
    * $3,500: Although the line can be drawn, to reach that cost with the given constraints, the solution would be outside the feasible region when trying to minimize cost.

**Answers**

* **Optimal Solution:** Produce 20 units on machine X and 0 units on machine Y.
* **Feasible but Suboptimal Isocost Lines:** $2,700 and $3,000.
* **Feasible and Optimal Isocost Line:** The isocost line that crosses (20,0) with a cost of $1200.
* **Not Feasible Isocost Line:** $3,500.