Problem 29 - Algebra 2

Question

After a wave is created by a boat, the height of the wave can be modeled using \(y=\frac{1}{2} h+\frac{1}{2} h \sin \frac{2 \pi t}{P},\) where \(h\) is the maximum height of the wave in feet, \(P\) is the period in seconds, and \(t\) is the propagation of the wave in seconds. How many times over the first 10 seconds does the graph predict the wave to be one foot high?

Step-by-Step Solution

Answer

The wave is one foot high twice per cycle, repeating as per the defined period intervals within 10 seconds, assuming each cycle completes under larger assumptions of \(h = 2\).

Step 1: Understanding the Function

The wave height function is given by \(y=\frac{1}{2}h + \frac{1}{2}h \sin \frac{2\pi t}{P}\). This represents a sinusoidal wave where the height \(y\) depends on time \(t\). The function oscillates between 0 and \(h\). We need to find when \(y = 1\) over the first 10 seconds.

Step 2: Setting Up the Equation

To find when the wave is 1 foot high, set \(y = 1\) in the equation: \(1 = \frac{1}{2}h + \frac{1}{2}h \sin \frac{2\pi t}{P}\). Simplify to\(1 = \frac{1}{2}h (1 + \sin \frac{2\pi t}{P})\). Divide both sides by \(\frac{1}{2}h\): \(\frac{2}{h} = 1 + \sin \frac{2\pi t}{P}\). This implies \(\sin \frac{2\pi t}{P} = \frac{2}{h} - 1\).

Step 3: Solving for Time Condition

For \(\frac{2}{h} - 1\) to be within \([-1, 1]\) (range for \(\sin\) function), \(h\) must satisfy \(h \geq 2\). In this case, solve \(\sin\frac{2\pi t}{P} = \frac{2}{h} - 1\) under the condition \(h \geq 2\).

Step 4: Solving the Sine Equation

Assuming \(h = 2\) for simplicity (allows full sinusoidal cycle): \(\sin\frac{2\pi t}{P} = 0\). Solve for \(t\): \(\frac{2\pi t}{P} = n\pi\) where \(n\) is an integer. This occurs when \(t = \frac{nP}{2}\). Over the first 10 seconds, find all allowable \(n\) so that \(0 \leq \frac{nP}{2} \leq 10\).

Step 5: Periodic Solutions

Since each wave fully completes within its period and for \(h = 2\) the wave is 1 foot high twice per full period, calculate values of \(n\) such that \(t = \frac{nP}{2}\) is within 10 seconds. Without the exact period \(P\), generally, calculate based on resolving periodic conditions e.g., giving points when \(\frac{nP}{2} < 10\).

Step 6: Conclusion

Under the assumption of \(h = 2\) and typical full wave cycles, the wave is exactly one foot high twice per cycle over time, occurring twice each period. Without explicit \(P\), specific count overlaid as illustrative rather than absolute, pending entire periodic description. However, solution assumes it cycles within such described periodicity.

Key Concepts

Wave Modeling

Wave modeling is a way to represent the behavior of waves using mathematical functions. It helps in predicting the properties of waves, like height, frequency, and speed, over time and space.
In the context of the exercise, the wave created by a boat can be modeled with the function: \[y = \frac{1}{2} h + \frac{1}{2} h \sin \frac{2 \pi t}{P},\] where:

• \(h\) is the maximum height of the wave in feet.
• \(P\) is the period of the wave in seconds.
• \(t\) represents time.

This mathematical model allows us to capture the behavior of the wave over time, showing how the height changes, and can be useful for understanding or predicting the motion caused by other sources like wind or tides.

Period of Trigonometric Functions

The period of a trigonometric function is the length of time it takes for the function to complete one full cycle. In our wave equation, the period is denoted by \(P\).
The equation \[y = \frac{1}{2}h + \frac{1}{2}h \sin \frac{2\pi t}{P}\] describes a wave that repeats its pattern every \(P\) seconds.
Understanding the period is crucial for determining how often the wave reaches a certain height within a given time frame. For instance, to find out how many times the wave is one foot high in 10 seconds, knowing the period \(P\) will tell us how many cycles fit into that time. Each cycle typically has points where the wave hits heights dependent on its sinusoidal pattern.

Sinusoidal Functions

Sinusoidal functions are functions that graph as a sine or cosine wave, and they are used to model oscillating behaviors. These functions take the form:

• Sine function: \(y = A \sin(\omega t + \phi)\)
• Cosine function: \(y = A \cos(\omega t + \phi)\)

where:

• \(A\) is the amplitude, the peak value the function reaches.
• \(\omega\) is the angular frequency, relating to how often the function oscillates.
• \(\phi\) is the phase shift, determining where the wave starts.

In our specific case, the sinusoidal part of the function \(\sin \frac{2\pi t}{P}\) describes how the height of the wave changes over time \(t\). Sinusoidal functions are perfect for modeling waves because they provide a smooth, continuous oscillation that mimics natural wave motion. They form the backbone of wave modeling in various applications, from physics to engineering, and help predict how waves behave under different conditions.