the water level varies from 10 inches at low tide to 40 inches at high tide. Low tide occurs at 9:15 a.m. and high tide occurs at 3:30 p.m. What is a cosine function that models the variation above and below the water level as a function of time and hours since 9:15 a.m.?
Let t be the number of hours past 9:15 a.m. (so, for example, t = 6.25 represents 3:30 p.m.).
At low tide (9:15 a.m.), the water level is 10 inches above some reference level. At high tide (3:30 p.m.), the water level is 40 inches above the same reference level. Therefore, the amplitude of the cosine function is (40 - 10) / 2 = 15 inches.
The period of the tide cycle is 12 hours (from low tide to low tide or high tide to high tide), so the frequency of the cosine function is 2π/12 = π/6.
The cosine function that models the variation of the water level as a function of time is:
h(t) = 15 cos(π/6(t - 3/4)) + 25
Note that the phase shift is 3/4 instead of the usual 1/2 since high tide occurs 3.25 hours after low tide, not 6 hours. The vertical shift is 25 instead of 10 to adjust the reference level.
are you sure this is correct?
I apologize, I made an error. Here's the corrected solution:
Let t be the number of hours past 9:15 a.m. (so, for example, t = 6.25 represents 3:30 p.m.).
At low tide (9:15 a.m.), the water level is 10 inches above some reference level. At high tide (3:30 p.m.), the water level is 40 inches above the same reference level. Therefore, the amplitude of the cosine function is (40 - 10) / 2 = 15 inches.
The period of the tide cycle is 6.25 hours (from low tide to high tide), so the frequency of the cosine function is 2π/6.25.
The cosine function that models the variation of the water level as a function of time is:
h(t) = 15 cos(2π/6.25(t - 1.25)) + 25
Note that the phase shift is 1.25 instead of the usual 1/2 since high tide occurs 6.25/4 = 1.5625 hours after low tide, not 3 hours. The vertical shift is 25 instead of 10 to adjust the reference level.
are you sure that right?
I apologize, there is another correction, please find the updated solution below:
Let t be the number of hours past 9:15 a.m. (so, for example, t = 6.25 represents 3:30 p.m.).
At low tide (9:15 a.m.), the water level is 10 inches above some reference level. At high tide (3:30 p.m.), the water level is 40 inches above the same reference level. Therefore, the amplitude of the cosine function is (40 - 10) / 2 = 15 inches.
The period of the tide cycle is 6.25 hours (from low tide to high tide), so the frequency of the cosine function is 2π/6.25.
The cosine function that models the variation of the water level as a function of time is:
h(t) = 15 cos(2π/6.25(t - 3.75)) + 25
Note that the phase shift is 3.75 instead of the usual 1/2 since high tide occurs 6.25/4 = 1.5625 hours after low tide, not 3 hours. The vertical shift is 25 instead of 10 in order to adjust the reference level.
any other corrections?
No, there are no further corrections. The function h(t) = 15 cos(2π/6.25(t - 3.75)) + 25 models the variation of the water level as a function of time, where t is the number of hours past 9:15 a.m.
no words just the steps
Sure, here are the steps to find the cosine function that models the variation of water level:
Step 1: Find the amplitude of the function: the amplitude is (high tide - low tide) / 2 = (40 - 10) / 2 = 15.
Step 2: Find the period of the function: the period is the time between two consecutive high tides or low tides, which is 6.25 hours.
Step 3: Find the frequency of the function: the frequency is 2π divided by the period, which is 2π/6.25.
Step 4: Find the phase shift of the function: the phase shift is the amount of time that has elapsed between the reference time (9:15 a.m.) and the time of the first high tide, which is 6.25/4 = 1.5625 hours. Convert this to decimal hours to obtain the phase shift of 3.75.
Step 5: Write the cosine function in the form h(t) = A cos(ωt - φ) + k, where A is the amplitude, ω is the angular frequency (ω = 2πf, where f is the frequency), φ is the phase shift, and k is the vertical shift (or the reference level). Substitute the values found in previous steps to obtain the final function:
h(t) = 15 cos(2π/6.25(t - 3.75)) + 25