Problem 38 - Calculus

Question

The depth of water in a tank oscillates sinusoidally 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 for depth is \( y(t) = 1.5 \sin(\frac{\pi}{3} t) + 7 \).

Step 1: Identify Key Parameters

First, we need to identify key parameters from the problem. The minimum depth is 5.5 feet, the maximum depth is 8.5 feet, and the tank oscillates every 6 hours. These values will help us determine the amplitude, period, and vertical shift of the sinusoidal function.

Step 2: Calculate the Amplitude

The amplitude of a sinusoidal function is half the difference between the maximum and minimum values. Calculate it using the formula: \( \text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} \). In this case, the amplitude is \( \frac{8.5 - 5.5}{2} = 1.5 \text{ feet} \).

Step 3: Determine the Period

The period is the time it takes to complete one full cycle of the oscillation. Given the information, the period is 6 hours. Since one full sine cycle completes in \(2\pi\) radians, we write the equation for the period as: \( \frac{2\pi}{b} = 6 \), solving for \( b \) gives \( b = \frac{2\pi}{6} = \frac{\pi}{3} \).

Step 4: Find the Vertical Shift

The vertical shift is the average of the maximum and minimum values, calculated as \( \text{Vertical Shift} = \frac{\text{Maximum} + \text{Minimum}}{2} \). Thus, the vertical shift is \( \frac{8.5 + 5.5}{2} = 7 \text{ feet} \).

Step 5: Write the Sinusoidal Function

Using the amplitude, period, and vertical shift, you can construct the formula for depth as a function of time. The general formula is: \( y(t) = a \sin(bt) + c \). Substituting in our values we get the function: \( y(t) = 1.5 \sin(\frac{\pi}{3} t) + 7 \).

Key Concepts

Amplitude in Sinusoidal Functions

The amplitude is a key characteristic of sinusoidal functions, representing how far the wave oscillates from the midline or its equilibrium position. It measures the height of the wave's peaks and the depth of its troughs.

To calculate the amplitude, simply determine the difference between the maximum and minimum values of the wave and divide it by two. This gives us the extent of the wave's oscillation.

• Formula: \( \text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} \)
• Example: Given a maximum water depth of 8.5 feet and a minimum of 5.5 feet, the amplitude is \( \frac{8.5 - 5.5}{2} = 1.5 \text{ feet} \).

Amplitude defines the strength or intensity of the sinusoidal oscillation, and in practical terms, it tells us how high and how low the wave will go relative to its central line.

Understanding the Period of a Sine Wave

The period of a sinusoidal function is the duration it takes to complete one complete cycle of wave motion. This is often expressed in terms of time or angle, and it determines how frequently the wave pattern repeats itself.

In practical scenarios, the period can help identify how often phenomena like tides or machine cycles occur. To find the period in terms of radians per unit time, you can use the formula:

• Period formula: \( \frac{2\pi}{b} = \text{Period} \)
• If the period is 6 hours, then by solving \( \frac{2\pi}{b} = 6 \), we find \( b = \frac{\pi}{3} \).

Understanding the period gives us insight into the timing and predictability of sinusoidal behaviors.

Exploring Vertical Shifts in Sinusoidal Graphs

Vertical shift is the amount a sinusoidal function is moved up or down on a graph. This shift allows the wave to start from a different base height or average position without altering the overall wave shape.

To find the vertical shift, calculate the average of the maximum and minimum values of the wave:

• Vertical shift formula: \( \frac{\text{Maximum} + \text{Minimum}}{2} \)
• For our example, the vertical shift would be \( \frac{8.5 + 5.5}{2} = 7 \text{ feet} \).

A vertical shift provides context for the wave's positioning on a graph, indicating the central point around which the wave oscillates. This is crucial for aligning data so that it reflects real-world conditions.