The height of the tide in a given location on a given day is modelled using the sinusoidal function h(t)=5 sin (30(t-5))+7. a) What time is high tide? What time is low tide?
So you want the t value which produces the max and min of the function, that is,
h '(t) = 0
5cos30(t-5) (30) = 0
cos 30(t-5) = 0
we know cos π/2 = 0 and cos 3π/2 = 0
30(t-5) = π/2 or 30(t-5) = 3π/2
t = 5.052 or t = 5.157
You did not define t, so you will have to make sense out of the value of t above.
To determine the time of high tide and low tide, we need to analyze the sinusoidal function h(t).
The general form of a sinusoidal function is h(t) = A sin(B(t - C)) + D, where:
- A represents the amplitude of the function (the maximum height or depth of the tide),
- B represents the frequency (in this case, it affects how quickly the tides oscillate between high and low),
- C represents the phase shift (it determines the horizontal shift in the tides),
- D represents the vertical shift (it determines the average height of the tides).
Let's analyze the given function h(t) = 5 sin(30(t - 5)) + 7:
1) Amplitude (A): The amplitude of the function is 5, representing the maximum height or depth of the tides.
2) Frequency (B): The frequency is determined by the coefficient of (t - C) term. In this case, B = 30, so we have 30(t - 5). The frequency can be calculated as 2π divided by B: frequency = 2π / 30.
3) Phase Shift (C): The phase shift is determined by the value inside the parentheses after the t-term. In this case, C = 5, so we have (t - 5). The positive sign suggests a shift to the right.
4) Vertical Shift (D): The vertical shift is represented by the constant D, which is 7 in this case. It determines the average height of the tides.
Now, let's calculate the time of high tide and low tide.
High tide occurs when the tide is at its maximum height, which is at the sinusoidal peak. In the function h(t) = 5 sin(30(t - 5)) + 7, the maximum height is equal to the amplitude (A) plus the vertical shift (D). So, the height of the high tide is 5 + 7 = 12 units.
To find the time of high tide, we need to solve the equation 5 sin(30(t - 5)) + 7 = 12. Let's solve for t:
5 sin(30(t - 5)) + 7 = 12
5 sin(30(t - 5)) = 12 - 7
5 sin(30(t - 5)) = 5
sin(30(t - 5)) = 1
Since sin(θ) = 1 only when θ = π/2, we have:
30(t - 5) = π/2
t - 5 = π/2 / 30
t = π/60 + 5
Therefore, the high tide occurs at t = π/60 + 5.
Similarly, low tide occurs when the tide is at its minimum height, which is at the sinusoidal trough. In the function h(t) = 5 sin(30(t - 5)) + 7, the minimum height is equal to the amplitude (A) minus the vertical shift (D). So, the height of the low tide is 5 - 7 = -2 units.
To find the time of low tide, we solve the equation 5 sin(30(t - 5)) + 7 = -2. Let's solve for t:
5 sin(30(t - 5)) + 7 = -2
5 sin(30(t - 5)) = -2 - 7
5 sin(30(t - 5)) = -9
sin(30(t - 5)) = -9/5
Since sin(θ) = -9/5 does not have a valid solution (as it is outside the range [-1, 1]), it means there is no time at which the tide reaches the depth of -2 units (low tide).
In conclusion:
- The time of high tide is t = π/60 + 5.
- There is no low tide in this period for which the tide reaches a depth of -2 units.
To find the times of high tide and low tide, we need to determine the values of 't' that correspond to these points on the graph of the sinusoidal function.
For the given function h(t) = 5 sin(30(t-5)) + 7, we know that the general form of a sinusoidal function is h(t) = A sin(B(t-C)) + D, where:
- A represents the amplitude
- B determines the period
- C represents the horizontal shift
- D represents the vertical shift
Comparing this general form to the given function, we can identify the values:
- A = 5 (amplitude)
- B = 30 (frequency or 1/period)
- C = 5 (horizontal shift)
- D = 7 (vertical shift)
For a sine function, high tide occurs at the maximum value of the function, which is equal to D + |A|. In this case, the maximum value is 7 + 5 = 12.
To find the time of high tide:
Step 1: Set h(t) = 12 and solve for t.
12 = 5 sin(30(t-5)) + 7
Simplifying the equation: 5 sin(30(t-5)) = 5
sin(30(t-5)) = 1
Since the range of sine function is -1 ≤ sin(x) ≤ 1, when sin(x) = 1, the angle (x) is equal to π/2 or 90 degrees.
So, 30(t-5) = π/2
Solving for t:
t - 5 = π/60
t = π/60 + 5
t = 5.0523
Therefore, high tide occurs at approximately t = 5.0523 (or about 5:03 a.m.).
Now, to find the time of low tide:
Step 2: Set h(t) = 2D - h(t), where 2D is the minimum value of the function.
2D - h(t) = 2(7) - (5 sin(30(t-5)) + 7)
2(7) - 2 * 5 sin(30(t-5)) = 14 - 5 sin(30(t-5))
To find the time of low tide, set 14 - 5 sin(30(t-5)) = 0
5 sin(30(t-5)) = 14
sin(30(t-5)) = 14/5
Since the range of sine function is -1 ≤ sin(x) ≤ 1, there is no value of t that satisfies this equation. Therefore, there is no low tide for this particular model.
In conclusion, the time of high tide is approximately 5:03 a.m., and there is no low tide based on the given model.