A coastal town records the height of the tide every hour over a 24-hour period. The data shows that the tide height varies sinusoidally with a maximum height of 10 feet and a minimum height of 2 feet. The tide reaches its maximum height at 6 AM and its minimum height at 6 PM. Write a sinusoidal function that models the tide height as a function of time.

A Ferris wheel has a diameter of 50 meters and completes one full rotation every 5 minutes. The lowest point of the Ferris wheel is 2 meters above the ground. Write a sinusoidal function that models the height of a passenger above the ground as a function of time.

The average monthly temperature in a city varies sinusoidally over the year, with a maximum of 30°C in July and a minimum of -10°C in January. Write a sinusoidal function that models the average monthly temperature as a function of time, where $t$ is the number of months since January.

A population of rabbits in a forest varies sinusoidally with a maximum population of 500 rabbits in April and a minimum population of 100 rabbits in October. Write a sinusoidal function that models the rabbit population as a function of time, where $t$ is the number of months since January.

The daylight hours in a city vary sinusoidally throughout the year, with a maximum of 15 hours in June and a minimum of 9 hours in December. Write a sinusoidal function that models the number of daylight hours as a function of time, where $t$ is the number of months since January.

Given the data points representing the average monthly temperatures in a city over a year, describe the steps you would take to fit a sinusoidal model to this data.

Explain how you would determine if a sinusoidal model is a good fit for a given set of data points.

A sinusoidal model for daylight hours over a year is given by $D(t)=12+3\sin (\frac{2\pi }{365}(t-80))$. Explain what each parameter represents in the context of this model.

Given a sinusoidal model $y=A\sin (Bx+C)+D$, describe how you would adjust the parameters to better fit data that shows a higher peak and a lower trough than the current model.

A sinusoidal model $y=5\sin (\frac{\pi }{6}x-\frac{\pi }{3})+2$ is used to represent a dataset. If the data shows a periodicity of 24 units instead of 12, how would you modify the model?

A sound wave can be modeled by the function $y(t)=3\sin (2\pi t)$, where $y$ is the displacement and $t$ is the time in seconds. Determine the amplitude, period, and frequency of this sound wave.

The average monthly temperature in a city can be modeled by the function $T(t)=20+10\sin (\frac{\pi }{6}t-\frac{\pi }{3})$, where $T$ is the temperature in degrees Celsius and $t$ is the month of the year (with $t=1$ corresponding to January). Determine the amplitude, period, and phase shift of this temperature model.

A climate cycle can be modeled by the function $C(t)=15\cos (\frac{\pi }{4}t+\frac{\pi }{2})+25$, where $C$ is the climate index and $t$ is the time in years. Identify the amplitude, period, and vertical shift of this climate cycle.

A sinusoidal function modeling the daylight hours in a city is given by $D(t)=12+3\sin (\frac{\pi }{6}t-\frac{\pi }{2})$, where $D$ is the number of daylight hours and $t$ is the month of the year (with $t=1$ corresponding to January). Determine the maximum and minimum number of daylight hours.