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?
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To find the cosine function that models the variation in inches above and below the water level, we need to consider the amplitude, period, and phase shift of the cosine function.
The amplitude of the cosine function is half the difference between the highest and lowest points of the variation, which is (64 - 12) / 2 = 26 inches.
The period of the cosine function is the time it takes for the function to complete one full cycle, which is the difference in time between high tide and low tide, or 1:30 pm - 8 am = 5.5 hours.
The phase shift of the cosine function is the horizontal shift of the function, which is determined by the starting time of 8 am.
Putting this information together, 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:
y(t) = 26cos[(2π / 5.5)(t - 8)], where t is the time in hours since 8 am.
This function has an amplitude of 26 inches, a period of 5.5 hours, and a phase shift of 8 hours.
check
You say
y(t) = 26cos[(2π / 5.5)(t - 8)], where t is the time in hours since 8 am.
well if it is 8 am t = 0
y(t) = 26cos[(2π / 5.5)( - 8)]
= 26 cos -524 degrees = 26 * -0.96 is about 25 inches below mean sea level at low tide
pretty close
I made a calculation error in my previous response. Let's calculate the correct cosine function for the given scenario.
At 8 am, the water level is at its lowest point, which is 12 inches below the average. So, when t = 0, the function should equal 12:
y(0) = 12 = A + 26 * cos(0 + φ)
12 = A + 26 * cos(φ)
cos(φ) ≈ 0.423
φ ≈ 116.57 degrees
Therefore, the cosine function is:
y(t) = 26 * cos[2π / 5.5 (t - 116.57)], where t is the time in hours since 8 am.
Thank you for pointing out the error, and I appreciate your feedback.