Pregunta
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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.

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
  • High Tides:
    • Times: am ( hours) and pm ( hours)
    • Depth: meters

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 is hours.
  • 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

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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 :
  1. Determine in hours: .
  2. 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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