The depth of the water in a bay varies throughout the day with the tides. Suppose that we can model the depth of the water with the following function.h(t)= 20+5.5 cos 0.25t
In this equation, ht is the depth of the water in feet, and t is the time in hours.
Find the following. If necessary, round to the nearest hundredth.
Number of tidal cycles per hour :
Minimum depth of the water : feet
Time between consecutive high tides: hours
Number of tidal cycles per hour: Since one tidal cycle completes in 24 hours, and there are 2 high tides in a cycle, the number of tidal cycles per hour would be 2/24 = 1/12.
Minimum depth of the water: To find the minimum depth, we need to find the minimum value of the function h(t). The function is of the form h(t) = A + B * cos (Ct + D), where A = 20, B = 5.5, C = 0.25, and D = 0.
To find the minimum, we set the derivative of h(t) equal to zero and solve for t:
h'(t) = -B * C * sin(Ct + D) = 0
sin(Ct + D) = 0
Ct + D = 0
0.25t = -0
t = 0
So, the minimum depth occurs at t = 0, which is the initial time. Substituting this value into the function, we get:
h(t) = 20 + 5.5 * cos (0.25 * 0)
h(t) = 20 + 5.5 * cos(0)
h(t) = 20 + 5.5 * 1
h(t) = 20 + 5.5
h(t) = 25.5 feet
Therefore, the minimum depth of the water is 25.5 feet.
Time between consecutive high tides: The time between consecutive high tides is the period of the function, which is given by T = 2π/C. In this case, C = 0.25, so the time between consecutive high tides is:
T = 2π/0.25
T = 8π
Rounding to the nearest hundredth, the time between consecutive high tides is approximately 25.13 hours.
To find the number of tidal cycles per hour, we need to determine the frequency of the cosine function.
The general formula for the frequency of a cosine function is given by f = 2π / T, where T is the period of the function.
In this case, the period is the time between consecutive high tides.
To find the period, we need to solve the equation h(t) = 20 + 5.5 cos(0.25t) = 20.
20 + 5.5 cos(0.25t) = 20
5.5 cos(0.25t) = 0
cos(0.25t) = 0
The cosine function is equal to zero when its argument is π/2, 3π/2, 5π/2, etc.
0.25t = π/2, 3π/2, 5π/2, ...
t = (π/2) / 0.25, (3π/2) / 0.25, (5π/2) / 0.25, ...
Simplifying these values, we get:
t = 2π, 6π, 10π, ...
This means that the period T is equal to 4π, which is approximately 12.57.
Now, we can find the frequency f = 2π / T = 2π / 12.57 ≈ 0.5.
Therefore, there are approximately 0.5 tidal cycles per hour.
To find the minimum depth of the water, we need to determine the lowest value of the function h(t).
The minimum value of the cosine function is -1, so the minimum depth occurs when 0.25t = π, 3π, 5π, etc.
0.25t = π, 3π, 5π, ...
t = π / 0.25, 3π / 0.25, 5π / 0.25, ...
Simplifying these values, we get:
t = 4π, 12π, 20π, ...
The lowest value of t is 4π, which is approximately 12.57.
Substituting this value into the function, h(t) = 20 + 5.5 cos(0.25(4π)) ≈ 20 - 5.5 = 14.5.
Therefore, the minimum depth of the water is approximately 14.5 feet.
To find the time between consecutive high tides, we need to determine the time it takes for the cosine function to complete one full cycle, which is the period.
We found earlier that the period is approximately 12.57 hours.
Therefore, the time between consecutive high tides is approximately 12.57 hours.
To find the number of tidal cycles per hour, we need to determine the period of the function. The period is the time it takes for the function to complete one cycle. In this case, the function is h(t) = 20 + 5.5 cos(0.25t), where t is in hours.
The period of a cosine function is given by 2π divided by the coefficient of t. In this case, the coefficient of t is 0.25. So, the period is:
Period = 2π / 0.25 = 8π
Now, to find the number of tidal cycles per hour, we can convert the period to cycles per hour:
Cycles per hour = 1 / (Period / 2π) = 1 / (8π / 2π) = 1 / 4 = 0.25
So, there are 0.25 tidal cycles per hour.
To find the minimum depth of the water, we need to find the minimum value of the function h(t). Since the cosine function oscillates between -1 and 1, the minimum value of h(t) occurs when cos(0.25t) = -1. This happens when 0.25t = π, since cos(π) = -1. Solving for t, we have:
0.25t = π
t = π / 0.25
t ≈ 12.57
Now, substitute this value of t back into the function h(t) to find the minimum depth:
h(t) = 20 + 5.5 cos(0.25t)
h(12.57) = 20 + 5.5 cos(0.25 * 12.57)
h(12.57) ≈ 20 + 5.5 cos(3.14)
h(12.57) ≈ 20 + 5.5 (-1)
h(12.57) ≈ 20 - 5.5
h(12.57) ≈ 14.50
So, the minimum depth of the water is approximately 14.50 feet.
To find the time between consecutive high tides, we need to consider the period of the cosine function. A high tide occurs when the depth of the water is maximum. The cosine function reaches its maximum value of 1 when cos(0.25t) = 1. This happens when 0.25t = 0, because cos(0) = 1. Solving for t, we have:
0.25t = 0
t = 0 / 0.25
t = 0
So, at t = 0, there is a high tide. The time between consecutive high tides is equal to the period of the function. From the previous part, we found that the period is 8π. So, the time between consecutive high tides is approximately 8π hours.