Find start and end times: multi-step word problems

Learning Objectives Deconstruct complex word problems into a sequence of time-based events. Model time intervals using arithmetic and geometric sequences to find unknown durations. Set up and solve systems of equations where time is a primary variable. Apply logarithmic functions to solve for start or end times in exponential growth and decay scenarios. Accurately calculate start or end times by working forwards or backwards through multiple, dependent time intervals. Convert between various units of time (days, hours, minutes, seconds) within a single multi-step problem. Analyze and interpret solutions in the context of the original problem, including calendar dates and times of day. A space probe is on a multi-year mission to Jupiter. If we know its arrival date and the...

TermDefinitionExample Time Interval (Duration)The elapsed time between a start point (t_start) and an end point (t_end). In multi-step problems, the total duration is the sum of several smaller intervals.A project has three phases. Phase 1 takes 3 days, Phase 2 takes 5 days, and Phase 3 takes 2 days. The total duration is 3 + 5 + 2 = 10 days. Sequential DependencyWhen an event's start time depends on the end time of a previous event. This creates a chain of calculations.Task B cannot start until Task A is complete. If Task A starts at 9:00 AM and takes 90 minutes, the earliest start time for Task B is 10:30 AM. Arithmetic Time ProgressionA series of time intervals where each subsequent interval changes by a constant amount (the common difference, d).A runner's training laps take...

Total Duration Formula t_{end} = t_{start} + \sum_{i=1}^{n} \Delta t_i The end time is the start time plus the sum of all individual time intervals (Δt_i). This can be rearranged to find the start time: t_{start} = t_{end} - \sum_{i=1}^{n} \Delta t_i. Arithmetic Sequence Sum (Total Duration) S_n = \frac{n}{2} (2a_1 + (n-1)d) Use this to find the total duration (S_n) of 'n' tasks whose individual durations form an arithmetic sequence, where 'a_1' is the duration of the first task and 'd' is the common difference. Exponential Decay/Growth Formula A(t) = A_0 \cdot (r)^{t/P} Models a quantity A(t) at time 't', starting with an initial quantity A_0. 'r' is the rate factor (e.g., 1/2 for half-life), and 'P' is the...