Sure! Let's calculate the correct cosine function step by step:
1. Period:
The given information tells us that low tide occurs at 8 am, and high tide occurs at 1:30 pm, which is 5.5 hours later. The period of a cosine function is the length of one cycle, which is 5.5 hours in this case. Therefore, the period (T) is 5.5.
2. Vertical shift:
As calculated earlier, the average of the high tide and low tide levels is 38 inches. This will be the vertical shift (midline) of the cosine function.
3. Phase shift:
The cosine function should be shifted to the right by 8 hours so that it aligns with low tide at 8 am. The phase shift (C) is 8.
Putting it all together, the corrected 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) = 26 cos((2π/5.5)(t - 8)) + 38
This function correctly represents the water level variation with the given period, vertical shift, and phase shift.
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?
To calculate the vertical shift in the cosine function, we need to consider the average of the high tide and low tide levels. Given: Low tide level = 12 inches High tide level = 64 inches To find the average: (12 + 64) / 2 = 38 inches The vertical shift in a cosine function is determined by this average value, which represents the midline of the oscillation. Therefore, the vertical shift in the cosine function for the given tide levels is 38 inches.
To determine the horizontal shift of the cosine function representing the water level variation, we use the given information: Given: The phase shift of the cosine function is 8 units to the right, aligning with low tide at 8 am. The horizontal shift represents the displacement of the function horizontally, indicating the starting point of the wave. Since the cosine function is shifted 8 units to the right to align with low tide at 8 am, the horizontal shift is 8 units to the right. Therefore, the horizontal shift of the cosine function is 8 units to the right, which corresponds to aligning the function with low tide at 8 am.
Therefore, the answer is:
f (t) = 26 cos (2pi/11 (t-2.5)) +38
My teacher says Careful with your direction, period, and shift. Please show work.
Can you help me correct this?
1 answer