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.
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.?
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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.
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.
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
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
no words
1. Amplitude = (high tide - low tide) / 2 = (40 - 10) / 2 = 15
2. Period = time between two consecutive high tides or low tides = 6.25 hours
3. Frequency = 2π / period = 2π/6.25
4. Phase shift = time elapsed between reference time (9:15 a.m.) and first high tide = 6.25/4 = 1.5625 hours = 3.75 in decimal hours
5. Function = h(t) = 15 cos(2π/6.25(t - 3.75)) + 25
2. Period = time between two consecutive high tides or low tides = 6.25 hours
3. Frequency = 2π / period = 2π/6.25
4. Phase shift = time elapsed between reference time (9:15 a.m.) and first high tide = 6.25/4 = 1.5625 hours = 3.75 in decimal hours
5. Function = h(t) = 15 cos(2π/6.25(t - 3.75)) + 25
1 answer