Mathematics
Grade 11
15 min
Times of everyday events
Times of everyday events
What you'll learn
- Identify whether a given table of values or set of ordered pairs represents an inverse variation relationship with 80% accuracy.
- Solve for the constant of variation (k) in an inverse variation equation given a set of values for x and y, demonstrating the correct algebraic steps in at least 4 out of 5 problems.
- Write an inverse variation equation in the form y = k/x, given a description of the inverse relationship and the value of the constant of variation, with 100% accuracy.
- Apply inverse variation equations to solve real-world problems (e.g., work rate problems, pressure-volume problems), showing all work and arriving at the correct solution in at least 3 out of 4 scenarios.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Model periodic daily events, such as daylight hours or tides, using sinusoidal functions.
Determine the amplitude, period, phase shift, and vertical shift of an event from a given data set or graph.
Construct a mathematical equation to represent the timing of a cyclical event.
Use a sinusoidal model to predict the value of a variable at a specific time.
Model events occurring at regular intervals using arithmetic sequences.
Calculate the time of the nth occurrence of an event described by an arithmetic sequence.
Ever wondered how meteorologists can predict the exact time of sunrise for any day of the year, or how a smartphone app knows the time of the next high tide? ☀️🌊
In this lesson, we will explore how the advanced mathematical tools you're lea...
2
Key Concepts & Vocabulary
TermDefinitionExample
Periodic FunctionA function that repeats its values at regular intervals or periods. The patterns of many everyday events, like the swing of a pendulum or the daily temperature cycle, can be described by periodic functions.The function describing the height of a point on a spinning Ferris wheel over time is periodic, as it repeats with every full rotation.
Sinusoidal ModelA mathematical model using a sine or cosine function to represent periodic data. It is defined by its amplitude, period, phase shift, and vertical shift.The number of daylight hours in a non-equatorial city over a year can be closely approximated by a sinusoidal model, peaking in summer and hitting a minimum in winter.
Amplitude (A)For a periodic event, the amplitude is half the distance between the...
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Core Formulas
General Sinusoidal Model
y = A \cos(B(t - C)) + D
This formula models a periodic event over time 't'. 'A' is the amplitude, 'D' is the vertical shift (average value), 'C' is the phase shift (start time of the cycle), and 'B' is related to the period. We often use cosine because its cycle naturally starts at a maximum, which is easy to identify in many real-world events (e.g., longest day of the year, highest tide).
Period Calculation
\text{Period} = \frac{2\pi}{|B|}
This formula connects the period of the event (how long one cycle takes) to the 'B' value in the sinusoidal model. When building a model, you often know the period and must solve for B using B = 2π / Period.
Arithmetic Sequence Term Formula
t_n = t...
4 more steps in this tutorial
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Sign Up Free to ContinueSample Practice Questions
Easy
In the general sinusoidal model for a daily event, y = A cos(B(t - C)) + D, what does the parameter D, the vertical shift, represent?
A.The time of the peak event
B.The average value of the event's data
C.The maximum displacement from the average
D.The duration of one full cycle
Easy
A city's daily high temperature fluctuates between a low of 15°C and a high of 25°C over a year. If this is modeled by a sinusoidal function, what is the amplitude (A)?
A.25°C
B.20°C
C.10°C
D.5°C
Easy
A train departs from a station at 7:00 AM, 7:12 AM, 7:24 AM, and so on. Which mathematical concept best describes the sequence of departure times?
A.sinusoidal model
B.logarithmic function
C.An arithmetic sequence
D.conic section
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What grade level is "Times of everyday events"?
Times of everyday events is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Times of everyday events?
You'll be able to: Identify whether a given table of values or set of ordered pairs represents an inverse variation relationship with 80% accuracy; Solve for the constant of variation (k) in an inverse variation equation given a set of values for….
Is "Times of everyday events" 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 Times of everyday events?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.