Let's solve **Question 5** step by step using linear programming. --- ## **Step-by-Step Solution (Problem 5: Laptop Manufacturer)** ### **Step 1: Define Variables** Let: - \( x \) = Number of laptops shipped to **Los Angeles** - \( y \) = Number of laptops shipped to **Boston** --- ### **Step 2: Write Constraints** #### **From the problem:** - Boston store needs at least **32 more laptops** than Los Angeles. - \( y \geq x + 32 \) - Los Angeles store needs at least **45 laptops**. - \( x \geq 45 \) - Must ship **at least 120 laptops in total**. - \( x + y \geq 120 \) --- ### **Step 3: Express Shipping Costs** - Shipping cost per laptop: - **To Boston**: $30 per laptop - **To Los Angeles**: $20 per laptop - **Objective function** (minimize total shipping cost): \[ \text{Cost} = 20x + 30y \] --- ### **Step 4: Graph the Feasible Region** Let's rewrite the constraints for graphing: - \( y \geq x + 32 \) - \( x \geq 45 \) - \( x + y \geq 120 \implies y \geq 120 - x \) - \( x \geq 0, \; y \geq 0 \) (Non-negativity) --- ### **Step 5: Find the Corner Points (Boundary Points)** Let's solve for the intersection points of the constraints: #### **a) Intersection of \( y = x + 32 \) and \( x + y = 120 \):** \[ y = x + 32 \\ x + y = 120 \implies x + (x + 32) = 120 \implies 2x = 88 \implies x = 44 \\ y = 44 + 32 = 76 \] So, **Point A: (44, 76)** #### **b) Intersection of \( x = 45 \) and \( y = x + 32 \):** \[ x = 45 \\ y = 45 + 32 = 77 \] So, **Point B: (45, 77)** #### **c) Intersection of \( x = 45 \) and \( x + y = 120 \):** \[ x = 45 \\ 45 + y = 120 \implies y = 75 \] So, **Point C: (45, 75)** --- ### **Step 6: Test Each Point Against All Constraints** - **Point A: (44, 76)** - \( x = 44 < 45 \Rightarrow \text{Not feasible} \) - **Point B: (45, 77)** - \( x = 45 \geq 45 \) - \( y = 77 \geq 45 + 32 = 77 \) - \( x + y = 45 + 77 = 122 \geq 120 \) - **Feasible** - **Point C: (45, 75)** - \( x = 45 \geq 45 \) - \( y = 75 \geq 45 + 32 = 77 \)? **No** (\( 75 < 77 \)) - **Not feasible** --- ### **Step 7: Evaluate the Objective Function at Feasible Points** We only have **Point B (45, 77)** as feasible. \[ \text{Cost} = 20x + 30y = 20(45) + 30(77) = 900 + 2310 = 3210 \] --- ## **Step 8: Final Answer** ### **Minimum shipping cost = \$3210** - **Laptops to Los Angeles:** 45 - **Laptops to Boston:** 77 --- ### **Summary Table** | Point | \(x\) | \(y\) | Feasible? | Cost (\$) | |--------|-------|-------|-----------|----------| | (44,76)| 44 | 76 | No | - | | (45,77)| 45 | 77 | Yes | 3210 | | (45,75)| 45 | 75 | No | - | --- ### **Final Answer:** > **Ship 45 laptops to Los Angeles and 77 to Boston for a minimum shipping cost of \$3210.** --- If you need the graph or want the solution for Problem 6, let me know!