Learning objective
When solving a problem involving periodic phenomena, we should consider what type of model will best represent a situation and whether we can verify that the model works as intended, by applying it to a problem situation and interpreting our results. When we make conclusions based on results, we should be prepared to report our findings.
Reporting with a model includes:
Information that is relevant to the audience
Enough detail that the audience understands the reasoning behind any recommendations
Reporting a model does not include:
Technical algebraic work
Mathematical jargon that could confuse the audience
12 seconds into an opera song, there is a section in which the singer must hold a note but alternate between high and low volume for 12 seconds. During this part of the song, the volume in decibels, v, is modeled by the following graph:
The volume after t seconds creates a sinusoidal pattern, with the following values recorded:
\text{Time in seconds, } t | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|
\text{Volume in decibels, } v | 82 | 88 | 82 | 76 | 82 | 88 | 82 |
Interpret the key features of the graph. What do they tell us about what's happening to the sound of the opera singer's voice?
Evodia was asked to find the volume of the opera singer's voice at 9.5 seconds, to the nearest decibel. She decided to extend the graph model:
Using Evodia's model, interpret the results of finding the volume at 9.5 seconds. Does Evodia's model provide a valid and sufficient answer to the problem? Why or why not?
Create a better model to find the volume of the singer's voice at 9.5 seconds, to the nearest decibel.
The tide rises and falls in a periodic manner which can be modeled by a trigonometric function. Volodymyr charts tide levels in order to determine when he can sail his ship into a bay and takes the following measurements:
Let t be the number of hours passed since low tide was first measured. The following table of values for the tide level, in d meters, models Volodymyr's charting:
\text{Time }(t \text{ in hours}) | 0 | 3 | 6 | 9 | 12 |
---|---|---|---|---|---|
\text{Tide level }(d \text{ in meters}) | 7 | 11 | 15 | 11 | 7 |
Volodymyr uses this data to build a function to model for the tide level: d \left(t \right)=-4 \cos \left(\dfrac{\pi}{6} t \right) + 11
Verify that the model is a good fit for the data.
The ship needs a depth of at least 11.7 \text{ m} to be able to sail in and out of the bay safely. Interpret the model to determine when Volodymyr can sail back into the bay.
Write a report with recommendations for Volodymyr's sailing excursions.
After working through the entire modeling cycle, which may include interpreting results and verifying our model, we generally need to report our findings. Some things to note when writing a report:
Reporting with a model includes:
Information that is relevant to the audience
Enough detail that the audience understands the reasoning behind any recommendations
Reporting a model does not include:
Technical algebraic work
Mathematical jargon that could confuse the audience