Problem 63 - Precalculus

Question

Refer to the following: The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula \(h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]\) where \(x\) is the number of hours since midnight and \(h\) is the height of the tide in feet. The average number of guests visiting the Magic Kingdom at Walt Disney World per day is given by \(n(x)=30,000+20,000 \sin \left[\frac{\pi}{2}(x+1)\right],\) where \(n\) is the number of guests and \(x\) is the month. If January corresponds to \(x=1\) how many people on average are visiting the Magic Kingdom per day in February?

Step-by-Step Solution

Answer

10,000 people visit per day in February.

Step 1: Understand the Problem

The function provided for the number of guests visiting the Magic Kingdom is \(n(x) = 30,000 + 20,000 \sin \left[\frac{\pi}{2}(x+1)\right]\). We need to determine the value of \(x\) for February and plug it into the function to find the average number of guests per day.

Step 2: Determine the Value for February

In the problem, it is mentioned that January corresponds to \(x=1\). Therefore, February will correspond to \(x=2\).

Step 3: Substitute the Value into the Function

Substitute \(x = 2\) into the formula: \[n(2) = 30,000 + 20,000 \sin \left(\frac{\pi}{2}(2+1)\right)\].

Step 4: Simplify the Expression Inside the Sine Function

Calculate the expression inside the sine function: \(\frac{\pi}{2}(2+1) = \frac{3\pi}{2}\).

Step 5: Evaluate the Sine Function

Find \(\sin\left(\frac{3\pi}{2}\right)\). The value of \(\sin\left(\frac{3\pi}{2}\right)\) is \(-1\).

Step 6: Calculate the Number of Guests

Substitute \(-1\) for the sine value: \[n(2) = 30,000 + 20,000(-1) = 30,000 - 20,000 = 10,000.\]

Step 7: Conclusion: State the Number of Guests

On average, 10,000 people visit the Magic Kingdom per day in February.

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool that helps to represent real-world situations using mathematical formulas and concepts. In our exercise, we are using mathematical models to understand how certain values change over time or in response to various factors. For example, in the harbor, the height of the tide and the number of visitors to the Magic Kingdom are modeled using sinusoidal functions. This is because both phenomena are naturally oscillating, with repeating patterns.

When creating a mathematical model, one needs to identify the core characteristics of the phenomenon being studied:

• Variables: The models use variables like `x` (time in hours or months) which affect the other variable `h` or `n` (height or number of visitors).
• Parameters: These are constant values like 5, 4.8, 30,000, and 20,000 in our models that help fit the model to the real-world data.
• Equations: Formulate equations that accurately describe the relationship between variables and parameters as done with sine functions in our examples.

Mathematical models simplify complex situations and provide us with predictions and insights, essential for planning and decision-making.

Periodic Functions

Periodic functions are mathematical functions that repeat their values at regular intervals. In our exercise, sinusoidal functions are used because they are a type of periodic function, specifically helpful for modeling waves and cycles, which naturally recur over time.

This periodic behavior is reflected in the function formulas like:

• For the tide: \( h(x) = 5 + 4.8 \sin \left[ \frac{\pi}{6}(x+4) \right] \), where the height of the tide repeats over a known cycle length (here influenced by \( \frac{\pi}{6} \)).
  
• For the guests: \( n(x) = 30,000 + 20,000 \sin \left[ \frac{\pi}{2}(x+1) \right] \), where the pattern of visitors changes cyclically over months, related to seasonal trends reflected by \( \frac{\pi}{2} \).

Important terms associated with periodic functions include:

• Amplitude: The height from the centerline to the peak, such as 4.8 feet in the tide model.
• Period: The length of one full cycle before the function repeats, influenced by coefficients like \( \frac{\pi}{6} \) or \( \frac{\pi}{2} \).
• Phase Shift: Any horizontal shift in the function, explained by the additions within the sine functions.

Understanding these aspects helps to interpret and predict behaviors described by periodic functions.

Real-World Applications

Trigonometric functions, especially periodic ones, have a vast range of real-world applications because of their ability to model repeating or oscillating processes accurately. In our examples, trigonometric functions capture the dynamics of tides in a harbor and guest attendance at a theme park.

Let's explore these scenarios further:

• Tide predictions: Modeling tides is crucial for navigation, coastal development, and recreation. Using sine functions, we can anticipate high and low tides to ensure safety and efficiency for boaters and builders.
• Visitor predictions: By understanding visitor trends in leisure spots like the Magic Kingdom, management can optimize staffing and resources, improving visitor experience and cost-effectiveness.

Trigonometric models thus extend beyond mathematics into fields such as meteorology, acoustics, and even economics. They allow foreseeing changes, preparing accordingly, and capitalizing on repetitive patterns. Understanding these applications highlights the practical importance of mastering trigonometric functions and periodic phenomena.