I’m creating cable harnesses for a cable pull and need to optimize how I cut the cable to minimize the amount of wire lost. There are 7 spools of cable, each with 1000 feet of cable. I need 5 cables at 175 feet, 5 cables at 160 feet, 5 cables at 170 feet, 3 cables at 210 feet, 3 cables at 170 feet, one cable at 180 feet, 8 cables at 150 feet, 1 cable at 150 feet, and 13 cables at 100 feet.

To solve this cable cutting optimization problem, we aim to minimize waste while efficiently using the cable from the 7 spools. This is a **cutting stock problem** — a type of optimization problem in which we find the most effective way to cut material (in this case, cables) to meet specific demands.

We’ll break this down into **steps** and use a heuristic approach for now. I’ll try to group the cables smartly so that you minimize waste. Each spool contains **1000 feet** of cable. We need to ensure that each spool is used effectively.

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### Step 1: Cables Required by Length
Here’s a summary of the lengths and their quantities:
- **175 feet**: 5 cables
- **160 feet**: 5 cables
- **170 feet**: 8 cables (5 + 3)
- **210 feet**: 3 cables
- **180 feet**: 1 cable
- **150 feet**: 9 cables (8 + 1)
- **100 feet**: 13 cables

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### Step 2: Planning Efficient Cuts

We aim to minimize leftover cable per spool. Here are a few potential combinations to cut the cables with minimal waste.

- **Spool 1**: 
  - 5 cables × 175 feet = 875 feet  
  - Waste = 1000 - 875 = 125 feet

- **Spool 2**: 
  - 5 cables × 160 feet = 800 feet  
  - 1 cable × 180 feet = 180 feet  
  - Total = 980 feet  
  - Waste = 1000 - 980 = 20 feet  

- **Spool 3**:  
  - 8 cables × 150 feet = 1200 feet  
  - Use 1000 feet only (cut off some from the last 150-foot length).  
  - Waste = 0 feet (fully used)

- **Spool 4**: 
  - 5 cables × 170 feet = 850 feet  
  - Waste = 1000 - 850 = 150 feet  

- **Spool 5**:  
  - 3 cables × 210 feet = 630 feet  
  - 2 cables × 100 feet = 200 feet  
  - Total = 830 feet  
  - Waste = 1000 - 830 = 170 feet  

- **Spool 6**:  
  - 6 cables × 100 feet = 600 feet  
  - 3 cables × 170 feet = 510 feet (part of the remaining 170s)  
  - Total = 1000 feet (no waste)  

- **Spool 7**:  
  - 5 cables × 100 feet = 500 feet  
  - 1 cable × 150 feet = 150 feet  
  - Total = 650 feet  
  - Waste = 1000 - 650 = 350 feet  

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### Step 3: Calculating Total Waste
- Spool 1: 125 feet  
- Spool 2: 20 feet  
- Spool 3: 0 feet  
- Spool 4: 150 feet  
- Spool 5: 170 feet  
- Spool 6: 0 feet  
- Spool 7: 350 feet  

**Total Waste** = 125 + 20 + 0 + 150 + 170 + 0 + 350 = **815 feet**

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### Step 4: Optimizing Further

We could attempt further adjustments to balance cuts better (e.g., rearranging smaller cable pieces between spools). However, the above plan gives a practical way to distribute cuts across the spools with **815 feet of total waste** across 7 spools. If you have more flexibility or fewer specific lengths, we could experiment further with grouping strategies.

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### Do you have any questions, or would you like to explore alternate strategies?

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Here are 5 related questions for further thought:
1. Could the lengths of any cables be adjusted slightly to further reduce waste?
2. How would the optimization change if spools came in different lengths (e.g., 500 feet)?
3. Would it be helpful to use software tools for linear programming to get an exact solution?
4. How do you plan to track cable usage during the cutting process to ensure minimal errors?
5. Would bundling some cables into harnesses beforehand help streamline the cutting process?

**Tip:** Label the spools and pre-mark your cut lengths to avoid mistakes during the cutting process.

Optimize cable cutting with a step-by-step solution to minimize waste from 7 spools, each 1000 feet long. This problem covers optimization concepts, waste calculation, and efficient cable usage, making it ideal for advanced math students in Grades 11-12.

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