Mathematics
Grade 11
15 min
Find start and end times: multi-step word problems
Find start and end times: multi-step word problems
What you'll learn
- Identify the standard form equation of a hyperbola (horizontal or vertical) given its general form equation, and correctly extract the values of 'a' and 'b' with 100% accuracy.
- Calculate the distance from the center to each focus ('c') of a hyperbola using the equation c² = a² + b², demonstrating proficiency by solving at least 8 out of 10 problems correctly.
- Determine the coordinates of the foci of a hyperbola, given its center and the value of 'c', and accurately plot the foci on a coordinate plane for at least 4 out of 5 different hyperbola equations.
- Explain the relationship between the values of 'a', 'b', and 'c' in a hyperbola and how these values determine the location of the foci, providing a clear and concise explanation in writing that demonstrates understanding of the underlying mathematical principles.
Tutorial Preview
1
Introduction & Learning Objectives
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...
2
Key Concepts & Vocabulary
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...
3
Core Formulas
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...
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Easy
A three-stage process begins at 10:00 AM. Stage 1 takes 1 hour and 45 minutes. Stage 2 starts immediately after and takes 90 minutes. Stage 3 starts after Stage 2 and takes 30 minutes. What is the end time of the entire process?
A.13:45
B.14:15
C.14:00
D.13:15
Easy
A scientific experiment starts at 1:40 PM and runs for a total of 200 minutes. At what time does the experiment conclude?
A.3:40 PM
B.4:20 PM
C.5:00 PM
D.4:40 PM
Easy
A project is defined by a series of tasks where the duration of each task forms an arithmetic sequence. If the first task (a₁) takes 5 hours and the common difference (d) is 1.5 hours, what is the duration of the 4th task (a₄)?
A.9.5 hours
B.11 hours
C.8 hours
D.14 hours
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What grade level is "Find start and end times: multi-step word problems"?
Find start and end times: multi-step word problems is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Find start and end times: multi-step word problems?
You'll be able to: Identify the standard form equation of a hyperbola (horizontal or vertical) given its general form equation, and correctly extract the values of 'a' and 'b' with 100% accuracy; Calculate the distance from the center to each….
Is "Find start and end times: multi-step word problems" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Find start and end times: multi-step word problems?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.