Analyzing Water Level At A Pier Using Trigonometric Functions
Introduction
In the realm of mathematical modeling, periodic functions play a crucial role in describing phenomena that exhibit repetitive behavior. One such application lies in predicting the water level fluctuations at coastal locations, particularly near piers and docks. The ebb and flow of tides, governed by the gravitational forces of the moon and the sun, create a cyclical pattern that can be effectively represented using trigonometric functions like cosine. In this comprehensive analysis, we will delve into a specific model that describes the water level at a pier, represented by the function y = 2.5cos(2π/12.5 * x) + 12, where y denotes the water level in meters and x represents the time in hours since the last high tide. Our primary objective is to determine the time elapsed after which the water level reaches a specific point. This exploration will not only involve understanding the mathematical underpinnings of the model but also interpreting the practical implications of the results in a real-world context.
The function provided, y = 2.5cos(2π/12.5 * x) + 12, is a classic example of a sinusoidal function, characterized by its smooth, oscillating wave-like pattern. The cosine function, in particular, is well-suited for modeling periodic phenomena that start at a maximum value, such as the high tide in our scenario. The various parameters within the function dictate the specific characteristics of the wave. The amplitude, represented by the coefficient 2.5, determines the vertical distance between the crest (highest point) and the midline of the wave. In our case, the water level fluctuates 2.5 meters above and below the midline. The period, which is the time it takes for one complete cycle of the wave, is determined by the term 2π/12.5. The period is calculated as 12.5 hours, indicating the time between successive high tides or low tides. The vertical shift, represented by the constant term 12, determines the midline of the wave. In this context, the midline represents the average water level at the pier. By carefully analyzing these parameters, we can gain a thorough understanding of the tidal patterns at the pier and accurately predict water level changes over time.
Understanding the behavior of this water level function is crucial for various practical applications. Mariners and boaters rely on accurate water level predictions to navigate safely, especially in areas with shallow waters or narrow channels. Port authorities and harbor masters use water level data to schedule vessel traffic and manage port operations efficiently. Coastal engineers and researchers utilize such models to design and maintain coastal structures, such as seawalls and breakwaters, ensuring their resilience against tidal forces. Furthermore, environmental scientists and ecologists employ water level data to study coastal ecosystems and understand the impact of tidal fluctuations on marine life and habitats. In essence, the ability to accurately model and predict water levels is essential for a wide range of activities, from navigation and infrastructure management to environmental conservation and scientific research. The insights gained from these models contribute to the safety, efficiency, and sustainability of coastal activities and resource management.
Problem Statement and Solution
Let's consider a specific scenario: at what time after high tide does the water level drop to 11 meters? To solve this problem, we need to substitute y = 11 into our equation and solve for x. This involves algebraic manipulation and a good understanding of trigonometric functions and their inverses. The equation becomes:
11 = 2.5cos(2π/12.5 * x) + 12
First, we isolate the cosine term by subtracting 12 from both sides:
-1 = 2.5cos(2π/12.5 * x)
Next, we divide both sides by 2.5:
-0.4 = cos(2π/12.5 * x)
Now, we need to find the angle whose cosine is -0.4. This is where the inverse cosine function, denoted as arccos or cos⁻¹, comes into play. Applying the inverse cosine to both sides, we get:
arccos(-0.4) = 2π/12.5 * x
Using a calculator, we find that arccos(-0.4) ≈ 1.9823 radians. However, it's crucial to remember that the cosine function is periodic, meaning it repeats its values at regular intervals. The cosine function has a period of 2π, which means that there are multiple angles that have the same cosine value. In the interval [0, 2π], there are two angles whose cosine is -0.4. One angle is in the second quadrant (between π/2 and π), which we found to be approximately 1.9823 radians. The other angle is in the third quadrant (between π and 3π/2) and can be found by subtracting 1.9823 from 2π, which gives us approximately 4.3009 radians. Therefore, we have two possible solutions for the equation:
- 9823 = 2π/12.5 * x
- 3009 = 2π/12.5 * x
To solve for x in each case, we multiply both sides by 12.5 and divide by 2π:
x = (1.9823 * 12.5) / (2π) ≈ 3.945 hours
x = (4.3009 * 12.5) / (2π) ≈ 8.555 hours
These two solutions represent the times after high tide when the water level first reaches 11 meters (approximately 3.945 hours) and when it reaches 11 meters again as the tide continues to fall (approximately 8.555 hours). In practical terms, this means that after a high tide, the water level will drop to 11 meters in about 3 hours and 57 minutes, and then again in about 8 hours and 33 minutes. This information is crucial for planning activities around the pier, such as docking boats or conducting maintenance work.
Interpretation and Validation
The solutions we obtained, approximately 3.945 hours and 8.555 hours, provide valuable insights into the tidal behavior at the pier. The first solution, 3.945 hours, indicates the time it takes for the water level to drop to 11 meters after a high tide. This is particularly useful for individuals planning activities that require a certain water depth, such as boaters who need to navigate shallow waters. By knowing the time when the water level will be at or above 11 meters, they can safely plan their journeys and avoid grounding their vessels. The second solution, 8.555 hours, represents the time when the water level reaches 11 meters again as the tide continues to recede. This information is equally important, as it allows for the prediction of low tide conditions and the planning of activities that are best suited for lower water levels, such as maintenance work on the pier or shoreline exploration.
To validate our solutions, we can graphically analyze the function y = 2.5cos(2π/12.5 * x) + 12. By plotting the function over time, we can visually confirm that the water level reaches 11 meters at approximately 3.945 hours and 8.555 hours after high tide. The graph will show a sinusoidal wave oscillating around the midline of 12 meters, with peaks representing high tides and troughs representing low tides. The points where the wave intersects the horizontal line y = 11 correspond to the solutions we calculated. This graphical validation provides a visual confirmation of our algebraic solutions and reinforces our understanding of the tidal patterns at the pier.
Furthermore, we can consider the physical context of the problem to assess the reasonableness of our results. The amplitude of 2.5 meters indicates a moderate tidal range, meaning the difference between high tide and low tide is approximately 5 meters. The period of 12.5 hours is consistent with the typical semi-diurnal tidal cycle observed in many coastal regions, where there are two high tides and two low tides per day. Given these physical parameters, the water level dropping to 11 meters within a few hours after high tide seems plausible. If our solutions were significantly different, such as several days or even weeks, it would raise concerns about the accuracy of our calculations or the validity of the model itself. By combining mathematical analysis with physical intuition, we can develop a robust understanding of the tidal dynamics at the pier and make reliable predictions about water level fluctuations.
Conclusion
In summary, the function y = 2.5cos(2π/12.5 * x) + 12 provides a valuable tool for modeling the water level at a pier. By solving the equation 11 = 2.5cos(2π/12.5 * x) + 12, we determined that the water level reaches 11 meters approximately 3.945 hours and 8.555 hours after high tide. These solutions have practical implications for various activities, including navigation, port operations, and coastal management. The ability to accurately predict water level fluctuations is essential for ensuring the safety, efficiency, and sustainability of coastal activities.
This analysis highlights the power of mathematical modeling in understanding and predicting real-world phenomena. By applying trigonometric functions and algebraic techniques, we can gain valuable insights into the complex dynamics of tidal systems. The water level model we explored serves as a prime example of how mathematical tools can be used to inform decision-making and improve the management of coastal resources. Furthermore, this example underscores the importance of understanding the physical context of a problem and validating mathematical solutions against real-world observations. By combining mathematical rigor with practical intuition, we can develop a comprehensive understanding of the world around us and make informed decisions that benefit society and the environment.
