The water level varies from 12 inches at low tide to 52 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 in inches above and below the water level as a function of time in hours since 9:15 a.m.?
We can use the cosine function to model the variation in inches above and below the water level as a function of time. The general form of a cosine function is:
y = A cos(Bx - C) + D
where:
A = amplitude (half the distance between the highest and lowest points of the function)
B = period (the time it takes for one complete cycle of the function)
C = horizontal shift (the phase shift of the function)
D = vertical shift (the average or midpoint of the function)
In this case, we know that the water level varies from 12 inches at low tide to 52 inches at high tide. Therefore, the amplitude is:
A = (52 - 12) / 2 = 20 inches
We also know that the period of the function is 6 hours, since there are approximately 6 hours between low tide and high tide. Therefore:
B = 2π / 6 = π/3
The horizontal shift is the time difference between the start of the function (9:15 a.m.) and the beginning of the cycle (low tide). Since low tide occurs at 9:15 a.m., there is no horizontal shift:
C = 0
Finally, the vertical shift is the average of the highest and lowest points of the function:
D = (52 + 12) / 2 = 32 inches
Putting this all together, we get the cosine function:
y = 20 cos(π/3 x) + 32
where x is the time in hours since 9:15 a.m. To find the water level at a specific time, simply plug in the value of x into the function and evaluate. For example, to find the water level at 1:30 p.m. (4.25 hours after low tide), we have:
y = 20 cos(π/3 x) + 32
y = 20 cos(π/3 x) + 32
y = 20 cos(π/3 * 4.25) + 32
y ≈ 41.3 inches
So the water level is approximately 41.3 inches above sea level at 1:30 p.m.
from the data:
difference between high tide and low tide
= 15:30 - 9:15 = 3:15
so the period = 6:30 or 6.5 hours
the bot equation would make it as 6 hours
period = 2π / k
6.5 = 2π/k
k = 2π/6.5 = 4π/13
so the equation would be:
y = 20 cos (4π/13 t) + 32
graph this using:
desmos.co m/calculator , to see that it is correct
(of course after cutting and pasting desmos.co m/calculator
you know you have to delete the space to fool this webpage)
Yes, you are correct that the period is 6.5 hours, not 6 hours as initially assumed. Therefore, the value of k in the cosine function should be 2π/6.5, which simplifies to 4π/13.
The correct equation for the water level as a function of time in hours since 9:15 a.m. is:
y = 20 cos (4π/13 t) + 32
Thank you for catching that mistake, and using Desmos.com/calculator is a great way to visually confirm the equation and its behavior.
To model the variation in inches above and below the water level as a function of time in hours since 9:15 a.m., we can use a cosine function. A cosine function represents periodic oscillations, which is appropriate for modeling the tide.
The general form of a cosine function is given by:
y = A * cos(B(x - C)) + D
In this case, A represents the amplitude of the tide (which is half the difference between the maximum and minimum values), B represents the frequency or how quickly the tide oscillates, C represents the phase shift or the time when the tide starts, and D represents the vertical shift or the average water level.
From the given information, we can determine these values:
Amplitude (A) = (52 - 12)/2 = 20
Frequency (B) = 2π/12 (since there are 12 hours between low tide and high tide in a period)
Phase Shift (C) = 3.25 hours (since high tide occurs at 3:30 p.m., which is 3.25 hours after 9:15 a.m.)
Vertical Shift (D) = (52 + 12)/2 = 32
Therefore, the cosine function that models the variation in inches above and below the water level is:
y = 20 * cos((2π/12)(x - 3.25)) + 32