Pregunta
The depth of the water at the end of a pier changes periodically along with the movement of tides. On a
particular day, low tides occur at
am and
, with a depth of 2.5 m , while high tides occur at
am and
, with a depth of 5.5 m . Let
be
.
Write a cosine model,
, for the depth as a function of time.
This amplitude is 1.5
The period is meters.
hours.
particular day, low tides occur at
Write a cosine model,
This amplitude is 1.5
The period is meters.
Ask by Potter Savage. in the United States
Jan 30,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The cosine model for the depth
as a function of time
is:
The value of
is
radians per hour, which is approximately
radians per hour.
Solución
To model the depth of the water as a function of time using a cosine function, we’ll follow these steps:
1. Understanding the Given Data
-
Low Tides:
- Times:
am ( hours) and pm ( hours) - Depth:
meters
- Times:
-
High Tides:
- Times:
am ( hours) and pm ( hours) - Depth:
meters
- Times:
2. Determining the Amplitude (
) and Vertical Shift (
)
-
Amplitude (
): -
Vertical Shift (
):
3. Determining the Period (
) and Angular Frequency (
)
-
Period (
):
The time between two consecutive low tides ishours. -
Angular Frequency (
):
4. Phase Shift Consideration
Since the first low tide occurs at
and cosine functions typically start at a maximum, we need to adjust the phase to start at a minimum. This can be done by introducing a negative amplitude:
5. Final Cosine Model
Putting it all together:
Summary
- Amplitude (
): meters - Vertical Shift (
): meters - Angular Frequency (
): radians per hour
Final Model:
Answer to the Specific Question
The value of
in the model is:
Numerically:
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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Mind Expander
To build your cosine model, let’s first clarify the parameters. The amplitude
is the distance between the midline and the maximum or minimum depth. Given
, that’s confirmed. The height difference between high tide (5.5 m) and low tide (2.5 m) gives a total variation of
, with the average (midline depth) being
.
Next, to find
which determines the period, note that one complete cycle of tide changes occurs from low tide at 12:00 am to the next low tide at 12:30 pm. This is a total of 12.5 hours for one complete cycle. The period
can be expressed as
. Therefore, to calculate
:
- Determine
in hours: . - Set
leading to .
Putting it all together, the cosine model is:
, where
is measured in hours.
Now, don’t forget to check your work for fun! Converting the period into a fraction gives you the ratio of the cycles—perfect for a group study session or a friendly math duel!
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