Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Tides are often measured by arbitrary height markings at some location. Suppose that a high tide occurs at noon when the water level is at 12 feet. Six hours later, a low tide with a water level of 5 feet occurs, and by midnight another high tide with a water level of 12 feet occurs. Assuming that the water level is periodic, use this information to find a formula that gives the water level as a function of time. Then use this function to approximate the water level at \(5: 30 \mathrm{P} . \mathrm{M}\).

Step-by-Step Solution

Answer

The water level at 5:30 PM is approximately 5.12 feet.

Step 1: Understanding the Problem

The water level follows a periodic pattern, reaching its maximum at noon and midnight (12 feet each) and its minimum at 6 PM (5 feet). We need to model this periodic behavior using a trigonometric function, typically a sine or cosine function.

Step 2: Define the Equation Form

The general form for a cosine function to model the water level is \[ L(t) = A \cos(B(t - C)) + D \] where:- \(A\) is the amplitude,- \(B\) determines the frequency,- \(C\) is the horizontal shift,- \(D\) is the vertical shift.

Step 3: Identify Amplitude and Vertical Shift

The amplitude \(A\) is half the distance between the maximum and minimum tide levels: \[ A = \frac{12 - 5}{2} = 3.5 \].The vertical shift \(D\) is the average of the maximum and minimum levels: \[ D = \frac{12 + 5}{2} = 8.5 \].

Step 4: Determine the Period

The time between successive high tides is 12 hours (from noon to midnight). For a cosine function, this corresponds to one full period, so \[ B = \frac{2\pi}{12} = \frac{\pi}{6} \].

Step 5: Horizontal Shift

Since the high tide (maximum level) occurs at noon (0 hours for convenience), this means there is no horizontal shift, hence \( C = 0 \).

Step 6: Construct the Equation

Substitute all known values into the cosine function:\[ L(t) = 3.5 \cos\left(\frac{\pi}{6}t\right) + 8.5 \].

Step 7: Substitute to Find Water Level at Specific Time

To find the water level at 5:30 PM, find the time elapsed since noon: 5.5 hours.Substitute \(t = 5.5\) into the equation:\[ L(5.5) = 3.5 \cos\left(\frac{\pi}{6} \times 5.5 \right) + 8.5 \].

Step 8: Calculate the Cosine Value

Calculate \( \frac{\pi}{6} \times 5.5 = \frac{11\pi}{12} \) and find the cosine value:\[ \cos\left(\frac{11\pi}{12}\right) = -\cos\left(\frac{\pi}{12}\right) \].Use a calculator to find \( \cos\left(\frac{\pi}{12}\right) \approx 0.9659 \), so \( -\cos\left(\frac{\pi}{12}\right) \approx -0.9659 \).

Step 9: Find the Water Level

Substitute the cosine value back into the equation:\[ L(5.5) = 3.5 \times -0.9659 + 8.5 \approx -3.38065 + 8.5 \approx 5.12 \].Thus, the water level at 5:30 PM is approximately 5.12 feet.

Key Concepts

Cosine Function

The cosine function is a type of trigonometric function, well-known for its periodic behavior. It is represented by the formula \( \,y = A \cos(Bx - C) + D\, \). In this formula:
\[\begin{align*} \text{- } & A \text{ is the amplitude, which influences the height of the wave from the middle line.} \ \text{- } & B \text{ changes the frequency or the length of the cycle, tightly packed or spread out.} \ \text{- } & C \text{ adjusts the horizontal shift.} \ \text{- } & D \text{ provides the vertical shift, moving the wave up or down on the graph.} \end{align*}\] The cosine function is especially useful in modeling waves and cyclical phenomena like tides, as it smoothly oscillates between a set maximum and minimum value.

Periodic Phenomena

Periodic phenomena are patterns or processes that repeat at regular intervals. This repetition can be observed in many natural and mechanical processes. For water levels, this means there's a predictable cycle of highs and lows. The concept of periodicity helps us anticipate events by examining the length of these cycles.
In mathematical terms, the period is defined as the time it takes for one complete cycle. For water levels or tides, this typically means the time from one high tide to the next. Knowing the period aids in setting the frequency \( B \) in our trigonometric model. In the given tide situation, your period is 12 hours (from noon to midnight), which makes the measurements easier and the predictions more accurate.

Water Level Modeling

Modeling water levels with trigonometric functions enables us to predict tides with accuracy. These models are advantageous because they are based on observable patterns in the data and described by sine and cosine functions, known for their oscillatory patterns.
By defining water levels using a cosine function, we can delicately reconstruct the rise and fall of tides over time. In the exercise, the cosine function has been chosen because it starts at a maximum, which conveniently matches with the high tide at noon. The formula is crafted by defining the amplitude, period, and vertical shifts that align with the real-world measurements of the water levels.

Amplitude and Vertical Shift

In trigonometric functions, the amplitude and vertical shift are crucial parameters that determine the height and average level of the modeled waves. - **Amplitude**: This is calculated as half the difference between the maximum and minimum values. In our tide example, the amplitude \( A \) becomes \( 3.5 \) feet. This measurement reflects how much the water level fluctuates from its average position.
- **Vertical Shift**: This represents the function's midline or the water's average level over time. For the tide problem, the vertical shift \( D \) is determined to be \( 8.5 \) feet, inferred from averaging 12 ft and 5 ft. These values help anchor our model, ensuring the highs and lows of our function reflect the actual peaks and dips of the tidal cycle. Together, these variables give us the overarching structure of our graph and sea-level prediction.