Problem 29 - Applied Calculus

Question

The depth of water in a tank oscillates once every 6 hours. If the smallest depth is 5.5 feet and the largest depth is 8.5 feet, find a possible formula for the depth in terms of time in hours.

Step-by-Step Solution

Answer

The formula is \(D(t) = 1.5 \cos\left( \frac{\pi}{3} t \right) + 7\).

Step 1: Understanding the Amplitude

First, identify the amplitude of the oscillation. The amplitude is half of the difference between the maximum and minimum depths. Here, the maximum depth is 8.5 feet and the minimum is 5.5 feet, so the amplitude is \((8.5 - 5.5) / 2 = 1.5\) feet.

Step 2: Identifying the Mean Depth

The mean depth is the average of the maximum and minimum depths. It is found by adding the maximum and minimum depths, then dividing by 2. So, the mean depth is \((8.5 + 5.5) / 2 = 7\) feet.

Step 3: Determining the Period

The period is the time it takes for one full cycle of oscillation to occur. According to the problem, the period is 6 hours. This is essential information for forming the sine or cosine function.

Step 4: Writing the General Formula

The formula to model the depth as a function of time will be a sine or cosine function. We have:\[ D(t) = A \sin\left( \frac{2\pi}{T} t \right) + C \]or \[ D(t) = A \cos\left( \frac{2\pi}{T} t \right) + C \]depending on the phase shift requirement.

Step 5: Applying Values to the Formula

Since the depth is at a maximum or minimum at the starting point, using cosine is appropriate. The formula becomes:\[ D(t) = 1.5 \cos\left( \frac{2\pi}{6} t \right) + 7 \] \[ D(t) = 1.5 \cos\left( \frac{\pi}{3} t \right) + 7 \]

Step 6: Verify the Formula

Plug in values for \(t\) to see if the formula gives correct maximum and minimum depths. For instance, at \(t = 0\), \(D(0) = 1.5 \cos(0) + 7 = 8.5\). At \(t = 3\), \(D(3) = 1.5 \cos\left( \frac{\pi}{3} \times 3 \right) + 7 = 5.5\). This confirms the formula works.

Key Concepts

Amplitude

Amplitude is a fundamental concept in the study of oscillating systems in calculus, particularly when dealing with trigonometric functions like cosine and sine. It measures how far the oscillation deviates from its central value, often seen in periodic functions.

The amplitude is calculated as half of the difference between the maximum and minimum values of the function. In mathematical terms, if you denote the maximum value as \(M\) and the minimum value as \(m\), the amplitude \(A\) is given by:

• \(A = \frac{M - m}{2}\)

Knowing the amplitude is crucial because it tells you how "tall" or "short" your oscillation is from the center line.

In the context of the exercise, the water depth in the tank oscillates between 8.5 feet and 5.5 feet. Calculating the amplitude, you get:

• \(A = \frac{8.5 - 5.5}{2} = 1.5\) feet

This indicates that the water level fluctuates by 1.5 feet above and below the mean depth. Understanding amplitude helps in graphing and describing wave phenomena, making it a key component in solving trigonometric problems.

Trigonometric Functions

Trigonometric functions are mathematical constructs that relate the angles of a triangle to its sides. They are widespread in calculus for modeling periodic phenomena, as they naturally represent oscillations, cycles, and waves.

The most frequently used trigonometric functions in these scenarios are sine (\(\sin\)) and cosine (\(\cos\)). Both are fundamentally related but shifted by a quarter cycle (or \(\pi/2\) radians). Here’s a quick overview of their characteristics:

• Periodicity: Both functions have a regular pattern that repeats over a fixed interval, known as the period.
• Range: The typical range for these functions is between -1 and 1, unless scaled by the amplitude.
• Phase Shift: This indicates a horizontal shift of the function along the x-axis and is essential when applying them to real-world scenarios.

To model real-world oscillations, such as the water levels in the exercise, you apply these functions using the general formula:

• For sine: \( y(t) = A \sin\left( \frac{2\pi}{T} t + \phi \right) + C \)
• For cosine: \( y(t) = A \cos\left( \frac{2\pi}{T} t + \phi \right) + C \)

where:

• \(A\) is amplitude,
• \(T\) is the period,
• \(\phi\) is the phase shift,
• \(C\) is the mean value around which the function oscillates.

Understanding these components enables you to precisely model any periodic function observed in nature or engineered systems.

Cosine Function

The cosine function is a specific type of trigonometric function that models periodic behavior commonly found in waveforms and oscillations. It is defined by the equation \(y = \cos(x)\), and its graph is a wave that repeats every \(2\pi\) radians.

In periodic modeling, the cosine function is particularly useful due to its distinctive starting point at \(x = 0\), where \(\cos(0) = 1\). This characteristic is leveraged in situations where a function needs to start from the maximum value, such as in the given exercise where the water depth reaches its peak at \(t = 0\).
Key features of the cosine function include:

• Period: The standard period is \(2\pi\), but it can be adjusted to other intervals \(T\) by scaling the input variable: \(\cos\left(\frac{2\pi}{T} t\right)\).
• Amplitude: The function can be scaled vertically by a factor given by the amplitude \(A\), which alters the height of the waves: \(A \cos(x)\).
• Vertical Shift: This is often added to change the midline of the oscillation: \(y = A \cos(x) + C\).

Using the cosine function can be particularly fitting when modeling systems that start at their highest point. For the water depth problem, the equation used is:
\[D(t) = 1.5 \cos\left(\frac{\pi}{3} t\right) + 7\]This equation effectively captures the periodic changes in water depth over a cycle of 6 hours, thanks to the cosine function's ability to naturally represent cyclical patterns.