Respuesta :
To create a cosine function that models the variation in inches above and below the water level as a function of time in hours since 8 am, we need to consider the amplitude, period, and phase shift of the cosine function.
Given:
- Low tide occurs at 8 am (time = 0 hours) with a water level of 12 inches.
- High tide occurs at 1:30 pm (time = 5.5 hours) with a water level of 64 inches.
- The range of variation in inches above and below the water level is from 12 inches to 64 inches.
The amplitude of the cosine function is half the difference between the maximum and minimum values, which is:
Amplitude = (64 - 12) / 2 = 26 inches
The period of the cosine function is the time it takes for one complete cycle, which can be determined by the difference in time between high tide and low tide, which is:
Period = 5.5 hours
The phase shift of the cosine function represents the horizontal shift of the graph. Since low tide occurs at 8 am (time = 0 hours), there is no phase shift.
Therefore, the cosine function that models the variation in inches above and below the water level as a function of time in hours since 8 am is:
f(t) = 26cos((2π/5.5)t)
where:
- f(t) is the variation in inches above and below the water level
- t is the time in hours since 8 am (low tide)
- 26 is the amplitude of the cosine function
- 2π/5.5 is the angular frequency corresponding to the period of 5.5 hours.
Given:
- Low tide occurs at 8 am (time = 0 hours) with a water level of 12 inches.
- High tide occurs at 1:30 pm (time = 5.5 hours) with a water level of 64 inches.
- The range of variation in inches above and below the water level is from 12 inches to 64 inches.
The amplitude of the cosine function is half the difference between the maximum and minimum values, which is:
Amplitude = (64 - 12) / 2 = 26 inches
The period of the cosine function is the time it takes for one complete cycle, which can be determined by the difference in time between high tide and low tide, which is:
Period = 5.5 hours
The phase shift of the cosine function represents the horizontal shift of the graph. Since low tide occurs at 8 am (time = 0 hours), there is no phase shift.
Therefore, the cosine function that models the variation in inches above and below the water level as a function of time in hours since 8 am is:
f(t) = 26cos((2π/5.5)t)
where:
- f(t) is the variation in inches above and below the water level
- t is the time in hours since 8 am (low tide)
- 26 is the amplitude of the cosine function
- 2π/5.5 is the angular frequency corresponding to the period of 5.5 hours.
