Water Level Calculation: Pier Function Explained

Hey everyone! Today, we're diving into a cool math problem about water levels at a pier. We'll be using a function to model the water's rise and fall, and figuring out when it reaches a specific height. Buckle up, it's going to be fun!

Understanding the Water Level Function

Okay, so we're given a function that describes the water level: $y = 2.5 \cos\left(\frac{2\pi}{12.5} x\right) + 12$. Now, let's break down what each part of this function means, because that's the key to understanding the whole problem, right?

• y: This represents the water level, and it's measured in meters. Think of it as how high or low the water is at any given time.
• x: This is the number of hours since the last high tide. High tide is when the water is at its highest point. So, x = 0 means we're at the peak of the water level.
• 2.5: This is the amplitude. It tells us how much the water level varies from the average. In this case, the water level goes 2.5 meters above and 2.5 meters below the average.
• \cos: This is the cosine function, which is a wave-like function. It's perfect for modeling the cyclical nature of tides – they go up and down in a regular pattern.
• \frac{2\pi}{12.5}: This part determines the period of the wave. The period is the time it takes for one complete cycle (from high tide to low tide and back to high tide). In this case, the period is 12.5 hours.
• + 12: This is the vertical shift. It tells us the average water level. The water level oscillates around 12 meters.

So, putting it all together, our function tells us how the water level (y) changes over time (x), following a cosine wave pattern with a period of 12.5 hours, an amplitude of 2.5 meters, and an average water level of 12 meters. This kind of function is super useful for predicting tide heights, which is important for stuff like boat navigation and coastal planning.

Now, let's get into the main question, because that's where the real fun begins. We're not just looking at the function, but solving it.

Solving for the Time: Water Level at 14 Meters

So, the big question is: After how many hours is the water level 14 meters? To figure this out, we need to use our function and solve for x when y = 14. Here's how we can do it:

1. Set up the equation: We start by setting the function equal to 14: $14 = 2.5 \cos\left(\frac{2\pi}{12.5} x\right) + 12$
2. Isolate the cosine function: First, subtract 12 from both sides: $2 = 2.5 \cos\left(\frac{2\pi}{12.5} x\right)$. Then, divide both sides by 2.5: $\frac{2}{2.5} = \cos\left(\frac{2\pi}{12.5} x\right)$, which simplifies to $0.8 = \cos\left(\frac{2\pi}{12.5} x\right)$.
3. Use the inverse cosine function: To get rid of the cosine, we use the inverse cosine (also known as arccosine), denoted as $\cos^{-1}$ or $\arccos$. Apply this to both sides: $\cos^{-1}(0.8) = \frac{2\pi}{12.5} x$. This gives us a value in radians.
4. Calculate the angle: Use a calculator to find $\cos^{-1}(0.8)$. Make sure your calculator is in radian mode! You'll get approximately 0.6435 radians. So, $0.6435 = \frac{2\pi}{12.5} x$.
5. Solve for x: Multiply both sides by 12.5 and divide by $2\pi$: $x = \frac{0.6435 \cdot 12.5}{2\pi}$. This gives us approximately 1.28 hours.

So, the water level will be 14 meters approximately 1.28 hours after the last high tide. But wait, there's more! Because the cosine function is a wave, the water level will reach 14 meters at other times during the tidal cycle. Let's see how to find those as well.

Finding All Solutions: Understanding the Wave's Behavior

The cosine function repeats itself. Because of this, our initial answer of 1.28 hours isn't the only time when the water level is 14 meters. There's another time during each cycle when the water reaches the same height. To find that, we need to understand the properties of the cosine wave.

The cosine wave is symmetrical. This means that if the water level reaches 14 meters at a certain time after high tide, it will reach the same height again at a corresponding time before the next low tide. To find this second time, we'll use the period of the wave (12.5 hours) and our first solution (1.28 hours).

The second time the water level is 14 meters can be found by taking the time it takes to get to the peak (high tide at x=0), adding the time to get to 14 meters (1.28 hours), and then subtracting that amount from the half-period. Here’s the formula:

• Half-period = Period / 2 = 12.5 / 2 = 6.25 hours.
• Second time = Half-period + (Half-period - 1.28 hours) = 6.25 + (6.25 - 1.28) = 11.22 hours.

So, the water level is also 14 meters at approximately 11.22 hours after the last high tide. And there we have it, the water level reaches 14 meters at approximately 1.28 hours and 11.22 hours after the last high tide. Awesome!

Generalizing the Solution

We found two solutions within one period (12.5 hours). Because the cosine function repeats, we can find additional solutions by adding multiples of the period to our initial solutions. If you need to find when the water level is 14 meters over a longer time frame (e.g., several days), you can calculate all the times as follows:

• Solution 1: $x = 1.28 + 12.5n$
• Solution 2: $x = 11.22 + 12.5n$

Where 'n' is an integer (0, 1, 2, -1, -2, and so on) representing the number of full cycles that have passed. This lets you find the times when the water level is 14 meters across any number of tidal cycles.

Practical Implications and Real-World Applications

This kind of water level analysis has real-world applications. Knowing when the water level reaches certain heights is essential for various activities:

• Navigation: Ships need sufficient water depth to navigate safely. Knowing the tide times and heights is critical for planning voyages and avoiding running aground.
• Coastal Construction: Building docks, piers, and other structures in coastal areas requires accurate tide predictions to ensure they are structurally sound and functional at all water levels.
• Recreation: Activities like boating, fishing, and surfing are heavily influenced by tide levels. Understanding tide charts helps people plan their activities for optimal conditions.
• Environmental Monitoring: Changes in tide patterns can indicate larger environmental changes, which is useful for conservation and management of coastal ecosystems.

Wrapping Up

We did it, guys! We broke down the water level function, solved for the time when the water level is 14 meters, and even found all possible solutions. We also got a glimpse of how important these calculations are in the real world. I hope you found this breakdown useful and interesting. Keep practicing, and you'll become a pro at these problems in no time. If you have any questions, feel free to ask. Cheers!