Elapsed time II

Learning Objectives Model elapsed time calculations using function notation. Apply modular arithmetic to solve problems involving cyclical time (e.g., clocks). Construct and evaluate functions that calculate end times across multiple days or time zones. Define the domain and range for time-based functions in a given context. Use piecewise functions to model scenarios where rates change over time. Analyze and interpret the results of time-based functions to solve complex real-world problems. A satellite needs to perform a maneuver exactly 100 hours from now. If it's currently 14:00 UTC on Tuesday, what time and day will the maneuver occur? 🛰️ Let's build a function to find out! This tutorial moves beyond simple time subtraction and addition. We will explore how to...

TermDefinitionExample Time as a FunctionA function that takes a start time and a duration as inputs and outputs a final time. It establishes a formal mathematical relationship between these quantities.Let T(start_hour, duration_hours) be the final hour. T(14, 5) = 19. This means starting at 14:00 and adding 5 hours results in 19:00. Modular ArithmeticA system of arithmetic for integers where numbers "wrap around" upon reaching a certain value, the modulus. This is the mathematical foundation of how clocks work.On a 12-hour clock, 8 + 5 = 13, which is 1 o'clock. In modular arithmetic, this is written as (8 + 5) mod 12 = 13 mod 12 = 1. Time in a Unified UnitConverting all time values into a single, smaller unit (like minutes or seconds from a reference point like midnight) to...

The Core Time Function (in a single unit) T_{final} = (T_{start} + \Delta T) \pmod{M} Use this formula when all times are converted to a single unit (e.g., minutes). T_start is the start time in minutes from midnight, ΔT is the duration in minutes, and M is the modulus (M=1440 for minutes in a 24-hour day). Day Change Calculation \text{Days Passed} = \lfloor \frac{T_{start} + \Delta T}{M} \rfloor This formula calculates how many full cycles (days) have passed. It uses the floor function (⌊x⌋), which rounds down to the nearest integer. T_start, ΔT, and M are in the same unit (e.g., minutes, with M=1440). Time Zone Conversion Function T_{local} = (T_{UTC} + \text{Offset}) \pmod{24} To find the local time, add the time zone offset to the UTC time and take the result mod...