The water level varies from 12 inches at low tide to 64 inches at high tide. Low tide occurs at 8 am and high tide occurs at 1:30 pm. What is 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?
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 is:
h(t) = 26cos(π/5 * (t - 8))
where:
- h(t) represents the height of the water level in inches above or below the average level at time t hours since 8 am
- π/5 is the frequency of the cosine function, which corresponds to the period of 10 hours (from low tide to high tide)
- t - 8 adjusts the starting point of the function to be at 8 am instead of 0
- The amplitude of the cosine function is 26, which is half of the difference between the high and low tide levels (64 - 12 = 52)
This cosine function will oscillate between -26 inches (12 - 26) at low tide and 26 inches (64 - 26) at high tide, reaching these extremes at 8 am and 1:30 pm, respectively.