The Martian Day 4

Focus: Piecewise functions
Objective: Model energy collection and usage with piecewise functions, graph the models, and analyze Watney’s power budget to survive 400 sols on Mars.
Scenario: Mark Watney relies on solar panels that generate power only during Martian daylight hours, with batteries storing excess energy for nighttime use. Energy demands vary based on habitat activities.
Math Skill: Create and graph piecewise functions to represent energy collection and consumption (CCSS.MATH.CONTENT.HSF.IF.C.7).

“If solar panels produce 1000 watt-hours during 12 hours of daylight and the habitat uses 800 watt-hours over 24 hours, what’s the net energy change in one sol?” Net is what is left over after the required amount is used.

• Daylight: 12 hours/sol, solar panels produce 1000 watt-hours total per sol (83.33 watt-hours/hour during daylight).

• Nighttime: 12 hours/sol, no solar production.

• Habitat energy use:

  • Normal operations: 33.33 watt-hours/hour (800 watt-hours/sol).

  • High-demand tasks (e.g., oxygenator repair): 50 watt-hours/hour (1200 watt-hours/sol).

• Battery capacity: 5000 watt-hours (initially full).

• Goal: Ensure energy sustainability for 400 sols.

How can Watney model his energy budget to maintain power over 24-hour periods?
To create the graphs for the lesson we need to visualize the piecewise functions for energy collection, energy consumption, and net energy over one Martian sol as well as the battery charge trend over multiple sols.

​So for a sol he collects 83.33 watt-hours/hour for an approximate total of
1000 watt-hours/sol

​normal operations ~ consume 800 watt-hours/sol

​high-demand operations consume 1200 watt-hours/sol

Normal operations:





High-demand tasks:

These piecewise functions represent the net energy rate (collection minus consumption, in watt-hours/hour). For normal operations, there’s a surplus during daylight (50 watt-hours/hour) and a deficit at night (-33.33 watt-hours/hour). For high-demand tasks, a smaller surplus during daylight (33.33 watt-hours/hour) and a larger deficit at night (-50 watt-hours/hour).

​These functions help Watney prioritize normal operations to maintain a sustainable energy supply and limit high-demand tasks to avoid battery depletion, ensuring power for critical systems over 400 sols.

​Recall that the battery capacity is 5000 watt-hours, so will the normal demand function continue to grow?

​Watney cannot sustain high-demand tasks long-term because they result in a net energy deficit of 200 watt-hours/sol (1000−1200). Starting with a 5000 watt-hour battery, continuous high-demand tasks would deplete it in 25 sols, far short of the 400 sols needed for survival. To survive, Watney must limit high-demand tasks, rely on normal operations (which provide a 200 watt-hours/sol surplus), or find additional energy sources.