Answered

Discover a wealth of information and get your questions answered on IDNLearn.com. Get prompt and accurate answers to your questions from our experts who are always ready to help.

The depth of the water at the end of a pier changes periodically with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm with a depth of 5.5 m. Let [tex]$t=0$[/tex] be 12:00 am.

Write a cosine model, [tex]$d = a \cos(bt) + k$[/tex], for the depth as a function of time.

1. This amplitude is [tex]$\square$[/tex] meters.
2. [tex]$a = $[/tex] [tex]$\square$[/tex]
3. Cone:

Sagot :

GinnyAnsweridn
Let's write a detailed step-by-step solution for determining the cosine model for the depth as a function of time.

### Step 1: Understand Given Information
- Low tide depth = 2.5 meters
- High tide depth = 5.5 meters
- Low tides occur at:
- 12:00 am (which corresponds to [tex]\( t = 0 \)[/tex] hours)
- 12:30 pm (which corresponds to [tex]\( t = 12.5 \)[/tex] hours)
- High tides occur at:
- 6:15 am (which corresponds to [tex]\( t = 6.25 \)[/tex] hours)
- 6:45 pm (which corresponds to [tex]\( t = 18.75 \)[/tex] hours)

### Step 2: Determine the Average Depth (Midline)
The average depth or midline, [tex]\( k \)[/tex], can be calculated as the average of the low tide depth and the high tide depth:
[tex]\[ k = \frac{\text{low tide depth} + \text{high tide depth}}{2} \][/tex]
[tex]\[ k = \frac{2.5 + 5.5}{2} = 4.0 \][/tex]
So, [tex]\( k = 4.0 \)[/tex] meters.

### Step 3: Determine the Amplitude
The amplitude, [tex]\( a \)[/tex], is half the difference between the high tide depth and the low tide depth:
[tex]\[ a = \frac{\text{high tide depth} - \text{low tide depth}}{2} \][/tex]
[tex]\[ a = \frac{5.5 - 2.5}{2} = 1.5 \][/tex]
So, the amplitude [tex]\( a \)[/tex] is 1.5 meters.

### Step 4: Determine the Period
The period of the tides is the time between two subsequent high tides (or low tides). In this case:
- Time difference between subsequent high tides:
[tex]\[ t_{\text{high,2}} - t_{\text{high,1}} = 18.75 - 6.25 = 12.5 \text{ hours} \][/tex]

So, the period [tex]\( P \)[/tex] is 12.5 hours.

### Step 5: Calculate the Value of [tex]\( b \)[/tex]
The value [tex]\( b \)[/tex] in the function [tex]\( d = a \cos(bt) + k \)[/tex] is calculated as:
[tex]\[ b = \frac{2\pi}{\text{period}} \][/tex]
[tex]\[ b = \frac{2\pi}{12.5} \approx 0.5026548245743669 \][/tex]

### Final Cosine Model
Given all the calculated values, we can now write the cosine model for the depth as a function of time [tex]\( t \)[/tex]:
[tex]\[ d(t) = a \cos(bt) + k \][/tex]
[tex]\[ d(t) = 1.5 \cos(0.5026548245743669 \cdot t) + 4 \][/tex]

### Summary
- Amplitude [tex]\( a \)[/tex] is: 1.5 meters
- [tex]\( b \approx 0.5026548245743669 \)[/tex]
- Average depth (midline) [tex]\( k \)[/tex] is: 4.0 meters

So, the finalized cosine model for the depth as a function of time is:
[tex]\[ d(t) = 1.5 \cos(0.5026548245743669 \cdot t) + 4.0 \][/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.