Problem 48 - Calculus

Question

For a boat to float in a tidal bay, the water must be at least 2.5 meters deep. The depth of water around the boat, \(d(t),\) in meters, where \(t\) is measured in hours since midnight, is $$d(t)=5+4.6 \sin (0.5 t)$$ (a) What is the period of the tides in hours? (b) If the boat leaves the bay at midday, what is the latest time it can return before the water becomes too shallow?

Step-by-Step Solution

Answer

(a) The period of the tides is \(4\pi\) hours. (b) The latest return time is approximately 1:26 AM the next day.

Step 1: Identifying Amplitude and Frequency

The given function is \( d(t) = 5 + 4.6 \sin(0.5t) \), which is a sinusoidal function. Here, the amplitude is 4.6, and the angular frequency is 0.5 rad/hr. These components are derived directly from the sinusoidal term \(4.6 \sin(0.5t)\).

Step 2: Calculate the Period

The period of a sinusoidal function \(d(t) = A \sin(Bt)\) is given by \( \frac{2\pi}{B} \), where \(B\) is the angular frequency. Here, \(B = 0.5\). Thus, the period is \( \frac{2\pi}{0.5} = 4\pi \) hours.

Step 3: Find when Depth is Below 2.5 m

Set \(d(t) = 2.5\) to find when the depth is too shallow. Rearrange the equation \(5 + 4.6 \sin(0.5t) = 2.5\) to \(4.6 \sin(0.5t) = -2.5\). Solve for \( \sin(0.5t) = \frac{-2.5}{4.6}\).

Step 4: Calculate the First Positive Time

Using a calculator, \( \sin^{-1}\left(\frac{-2.5}{4.6}\right) \approx -0.596\). Since sine is periodic with period \(2\pi\), \(0.5t = -0.596 + 2\pi k\) where \(k\) is an integer. The first positive solution \(t_1 = \frac{2\pi - 0.596}{0.5} \approx 12.87\) hours.

Step 5: Calculate the Next Time Interval

Since \(d(t)\) is periodic with period \(4\pi\), additional times when the water is shallow occur periodically. The next time increments by the period, \(t_1 + 4\pi \approx 12.87 + 12.57 \approx 25.44\) hours, which corresponds to the next day.

Step 6: Determine Latest Return Time

Since the boat leaves at midday (12:00 or 12.0 hours), it cannot return before the water reaches a depth of 2.5 meters again after midday, which corresponds to the next cycle at approximately \(25.44\) hours in the time cycle, or 1:26 AM the next day.

Key Concepts

Sinusoidal Functions

Sinusoidal functions are a type of trigonometric function that form the basis of many natural phenomena, such as sound waves and tides in the ocean. These functions are characterized by their smooth, repetitive oscillations. The general mathematical form of a sinusoidal function is given by:
\[d(t) = A \, \sin(Bt + C) + D\]
In this expression:

• \(A\) denotes the amplitude of the wave. It represents the peak value of the wave from its midline.
• The term \(B\) is related to the angular frequency, governing how quickly the oscillations occur.
• \(C\) is the phase shift, which determines where the function starts along the horizontal axis.
• \(D\) is the vertical shift, moving the entire function up or down along the vertical axis.

In the context of our exercise, the sinusoidal function is \(d(t) = 5 + 4.6 \sin(0.5t)\). The amplitude is 4.6, indicating the maximum deviation of the water level from its average. The vertical shift sets the average depth at 5 meters.

Periodic Functions

Periodic functions are mathematical functions that repeat their values at regular intervals, called periods. Sinusoidal functions are a classic example of periodic functions. A function is periodic if there exists a positive value \(T\) such that \(f(t+T) = f(t)\) for all values of \(t\). The smallest such \(T\) is the period of the function.
The formula for calculating the period of a sinusoidal function is \(\frac{2\pi}{B}\), where \(B\) represents the angular frequency:

• In our problem, \(B = 0.5\), so the period is \(\frac{2\pi}{0.5} = 4\pi\) hours.

This means the water depth function repeats its cycle every \(4\pi\) hours. Understanding the periodic nature is crucial for predicting when certain depth conditions, such as a safe boat return, will occur.

Angular Frequency

Angular frequency is a key component in understanding sinusoidal and periodic functions. It is denoted by the symbol \(B\) in the standard sinusoidal function formula, and it measures the rate of oscillation in radians per time unit.
Angular frequency is expressed as:
\[\omega = \frac{2\pi}{T}\]
or in terms of the function as the coefficient of \(t\) in \(\sin(Bt)\). It connects the concept of frequency (number of oscillations per time unit) with the period (how long it takes to complete one cycle):

• In our tidal problem, this value is \(0.5\), meaning the sine wave completes "half" a cycle every hour.

The angular frequency provides insights into how rapidly changes are occurring. High angular frequency implies rapid oscillations, while lower values indicate slower cycles. Understanding angular frequency aids in predicting system behavior over time, relevant for knowing tidal movements in bays.