On a typical day at an ocean port, the water has a maximum depth of 20m at 8:00AM. The minimum depth of 8m occurs 6.2h later. Assume that the relation between the depth of the water and time is a sinusoidal function.
write an equation for the depth of the water at any time, t hours
I got h=6cos(2pi((t-8)/12.4))+14
how do I get the equation for sin?
I got h=6sin(2pi((t-9.8)/12.4))+14
since the highest point is at 8:00AM, and calculated the lowest point would be at 1:48AM(1.8@x-axis). Then I did 1.8+8 = 9.8 for the phase shift
but the back of the book says
h=6sin(2pi((t-4.9)/12.4))+14
thanks in advance
If high tide is at 8 am
and the time from high to low is 6.2 hr
then half tide (mid-tide) where the sin function is zero is at 8 - 3.1 = 4.9 am
Oh right, it starts in the middle, thanks!
To find the correct equation for the depth of the water as a sinusoidal function, let's go through the process step by step:
1. Start with the general form of a sinusoidal function:
h = A*sin(B(t - C)) + D
- A: Amplitude (half the difference between the maximum and minimum values)
- B: Frequency (one full cycle is completed every 2π/B units of time)
- C: Phase shift (horizontal shift of the graph)
- D: Vertical shift (shift of the entire graph up or down)
2. Given information:
- Maximum depth: 20 m at 8:00 AM (t = 8)
- Minimum depth: 8 m occurs 6.2 hours later (t = 8 + 6.2 = 14.2)
3. Amplitude:
Amplitude = (20 - 8) / 2 = 6
4. Frequency:
One full cycle occurs in 6.2 hours, so the period is 6.2 hours.
Frequency = 2π / 6.2 = π / 3.1
5. Phase shift:
To determine the phase shift, we need to find when the minimum depth occurs (t = 14.2).
We subtract this time from the time of the maximum depth (t = 8) to get the phase shift.
Phase shift = 14.2 - 8 = 6.2
6. Vertical shift:
The water's depth oscillates between 8 m (minimum) and 20 m (maximum).
The average value between these two points is (8 + 20) / 2 = 14.
So, the equation for the depth of the water at any time, t hours, is:
h = 6*sin((π / 3.1)(t - 6.2)) + 14
However, it seems there might be an error in the back of the book you mentioned, as a phase shift of 4.9 would not align with the given times of maximum and minimum depths.
To find the equation for the depth of the water at any time, t hours, we need to understand the properties of a sinusoidal function. A sinusoidal function can be represented by either a sine or cosine function. The choice between sine and cosine depends on the specific situation and the given information.
In your case, you correctly assumed that the highest point occurs at 8:00 AM and found that the lowest point would be at 1:48 AM. However, your calculation for the phase shift seems to have a mistake.
Here's the correct way to find the equation for the depth of the water:
1. Determine the amplitude:
The amplitude is the vertical distance between the highest and lowest points. In this case, the highest point is 20m and the lowest point is 8m. Subtracting the lowest from the highest, we get an amplitude of 20 - 8 = 12.
2. Determine the period:
The period is the length of one complete cycle. In this case, it took 6.2 hours from the highest point to the lowest point. Therefore, the period is 6.2 hours.
3. Determine the phase shift:
The phase shift is the horizontal displacement of the wave. It represents the time shift from the starting point (highest point) of the wave. In this case, since the highest point occurs at 8:00 AM and the cycle takes 6.2 hours, we need to convert 8:00 AM to a decimal form. 8:00 AM is equivalent to 8 hours. To calculate the phase shift, subtract the period (6.2 hours) from the starting point (8 hours), giving us a phase shift of 1.8 hours.
Now, let's determine the correct equation using a sine function:
h = 6sin(2π((t - 1.8)/6.2)) + 14
The amplitude (A) is 6, the period (P) is 6.2, and the phase shift (C) is 1.8. The constant term (D) is 14, which represents the average depth of the water.
The equation you provided (h = 6sin(2π((t - 9.8)/12.4)) + 14) seems to have a miscalculation in the phase shift. Instead of subtracting the period from the starting point (8), you added it. This resulted in the phase shift being incorrectly calculated as 9.8.
The correct equation provided in the book (h = 6sin(2π((t - 4.9)/12.4)) + 14) has a phase shift of 4.9, which is half of the calculated period. This is because a sine function starts from its highest point (or middle) before reaching the lowest point.
Hope this clarifies the process involved in finding the equation for the depth of the water at any time using a sinusoidal function.