Fuel for the Frozen Trail
You’ve joined a planning team organizing an Arctic expedition. The crew will travel for 60 days across snow and ice, carrying everything they need—including all their food.
Each explorer is expected to do hard physical work pulling sleds, setting up camp, and staying warm in freezing conditions. Getting the right number of calories each day is essential to staying healthy and strong.
But food is heavy. The more you bring, the slower you move. The less you bring, the more you risk running low.
Expedition Parameters
- Duration: 60 days
- Team Size: 10 explorers
- Total Supplies: 500 kg (food and fuel)
- Maximum Load per Person: 50 kg
Calorie Scenarios
Scenario A: Overestimating Capability
- Daily Calories per Explorer: 1,800 cal/day
- Belief: Enough for expected activity
Scenario B: Underestimating Capability
- Daily Calories per Explorer: 2,200 cal/day
- Belief: Safer margin beyond estimated need
Your Challenge
Using your understanding of calculus, model the following:
🔹 1. Energy Balance Over Time
- How does daily calorie intake affect energy levels?
- Define a rate of change (derivative) to represent daily energy gain or loss.
🔹 2. Total Energy Debt or Surplus
- Integrate the daily energy imbalance over time.
- Represent the total energy deficit or surplus over multiple days.
🔹 3. Threshold Effects
- What happens if an explorer’s energy debt becomes too large?
- Define a critical threshold of performance decline.
🔹 4. Team Impact
- If an explorer drops out, how does redistributing their load affect others?
- Model how increased load raises caloric need, impacting remaining team members.
Hints and Considerations
- Assume the true calorie need per person is 2,000 cal/day.
- Use differential equations to model daily changes.
- Use integration to calculate total energy change over time.
- Consider a function linking load carried to calorie consumption (e.g., linear increase).
Optional Exploration
- Plot energy debt/surplus over time.
- Compare curve shapes in both scenarios.
- Analyze long-term consequences of overfeeding vs. underfeeding.
- Determine how the team’s dynamics change with load redistribution.