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 \(5: 30\) P.M.

Step-by-Step Solution

Answer

The water level at 5:30 P.M. is approximately 5.88 feet.

Step 1: Understand the Periodicity

The problem states that the tides are periodic. We have high tides at noon and midnight, both of which have a water level of 12 feet. The low tide occurs at 6 p.m. with a water level of 5 feet. So, the period of the tide cycle is 12 hours.

Step 2: Model the Tide as a Sine Function

We can model the water level using a sine function, since sine functions are naturally periodic. The general form of a sine function is \( f(t) = A \sin(Bt + C) + D \). Here, \( A \) is the amplitude, \( B \) relates to the period, \( C \) is the phase shift, and \( D \) is the vertical shift.

Step 3: Determine the Amplitude and Vertical Shift

The maximum water level is 12 feet and the minimum water level is 5 feet. The amplitude (\( A \)) is half the difference of the maximum and minimum 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: Calculate the Period and Frequency

Since the period is 12 hours, we use the relationship \( B = \frac{2\pi}{ ext{period}} \). Therefore, \( B = \frac{2\pi}{12} = \frac{\pi}{6} \).

Step 5: Determine the Phase Shift

At \( t = 0 \) (noon), the high tide corresponds to the maximum value of \( f(t) = 12 \). A sine function reaches its maximum at \( \frac{\pi}{2} + 2k\pi \). Therefore, solve \( \frac{\pi}{6} \times 0 + C = \frac{\pi}{2} \) for \( C \): \( C = \frac{\pi}{2} \).

Step 6: Write the Function

Combine all parts into the function: \( f(t) = 3.5\sin\left(\frac{\pi}{6}t + \frac{\pi}{2}\right) + 8.5 \).

Step 7: Approximate Water Level at 5:30 P.M.

5:30 P.M. is 5.5 hours from noon, so substitute \( t = 5.5 \) into the function: \( f(5.5) = 3.5\sin\left(\frac{\pi}{6}(5.5) + \frac{\pi}{2}\right) + 8.5 \). Calculating inside, \( 3.5\sin\left(\frac{\pi}{6} \cdot 5.5 + \frac{\pi}{2}\right)\approx 3.5\sin\left(2.875\right)\). Then calculate the value to find \( f(5.5) \approx 5.8809 \).

Key Concepts

Periodic Functions

When we say a function is periodic, it means that it repeats its values in regular intervals or periods. In the context of tide functions, the height of the tide recurs in cycles over a set duration, illustrating a predictable rise and fall.

• The given problem presents a scenario where the high tide occurs at noon and midnight, both at 12 feet, and the low tide at 6 P.M. is 5 feet. This denotes a 12-hour period cycle.
• Understanding the interval or period helps in predicting the occurrence of future high and low tides, and forms the basis of modeling the tide with mathematical functions.
• For any repeated natural phenomena like tides, periodic functions offer a valuable approach to anticipate behavior over time. This cyclic nature is essential for functions like sine waves.

Sine Functions

Sine functions are a fundamental type of periodic function. They are essential in modeling waves, cycles, or patterns found in nature, such as sound waves or tides.
The generic sine function can be written as:

• \( f(t) = A \sin(Bt + C) + D \)
• This equation helps us capture various characteristics of cycles through its parameters.
• In this context, we use the sine function to model tide patterns because it encompasses the periodic and smooth changes in tide height.
• The sine function starts at zero, reaches its maximum, returns to zero, hits its minimum, and finishes a cycle back at zero, which mimics the ebb and flow of tides over time.

Amplitude Calculation

Amplitude refers to how far a wave's maximum point is from its rest position. It signifies the strength or intensity of the wave.
For tide modeling:

• We calculate amplitude as half the difference between the maximum tide level and the minimum tide level.
• In our problem, the maximum is 12 feet and the minimum is 5 feet. Therefore, the amplitude \( A = \frac{12 - 5}{2} = 3.5 \) feet.
• This figure helps in constructing the sine wave of the tide function, indicating the extent of variation from the average tide level.

Vertical Shift

Vertical shift is the offset from the midline of the sine function, which tells us the function's change from its expected center, or zero level.
In tide problems:

• The vertical shift represents the average tide level and is calculated as the mean of the maximum and minimum tide heights.
• Here, it turns out to be \( D = \frac{12 + 5}{2} = 8.5 \) feet.
• This shifts the sine wave up or down in order to fit the average level of tides in the given location, ensuring that the sine function correctly portrays the tide cycle around "mean sea level."

Phase Shift

Phase shift involves offsetting the entire wave horizontally. This shift indicates at which point in the cycle our function starts.

• In our tidal model, the phase shift \( C \) is necessary because the sine function normally starts at zero, but our tides start at high tide.
• To find the phase shift, solve for \( C \) in the equation at high tide time \( t = 0 \). Knowing the sine function reaches its peak at \( \frac{\pi}{2} \), we set it up as \( \frac{\pi}{6} \times 0 + C = \frac{\pi}{2} \).
• This solves to \( C = \frac{\pi}{2} \), ensuring that the function accurately starts at the correct high tide point.

Phase shifting allows us to correctly align the modeled wave with real-world observations of tide occurrences.