Problem 75 - Algebra and Trigonometry

Question

Height of a Wave As a wave passes by an offshore piling, the height of the water is modeled by the function $$h(t)=3 \cos \left(\frac{\pi}{10} t\right)$$ where \(h(t)\) is the height in feet above mean sea level at time \(t\) seconds. (a) Find the period of the wave. (b) Find the wave height, that is, the vertical distance between the trough and the crest of the wave.

Step-by-Step Solution

Answer

(a) The period is 20 seconds. (b) The wave height is 6 feet.

Step 1: Understand the Cosine Function

The function given is \(h(t) = 3 \cos \left(\frac{\pi}{10} t\right)\). This is a cosine function of the form \(A \cos(Bt)\), where \(A\) is the amplitude and \(B\) affects the period of the function.

Step 2: Calculate the Period of the Wave

The period \(T\) of a cosine function \(A \cos(Bt)\) is given by the formula \(T = \frac{2\pi}{B}\). In this problem, \(B = \frac{\pi}{10}\). Thus, the period is \(T = \frac{2\pi}{\frac{\pi}{10}} = 20\). Therefore, the period of the wave is 20 seconds.

Step 3: Determine the Amplitude

The amplitude \(A\) of the function \(h(t) = 3 \cos \left(\frac{\pi}{10} t\right)\) is the coefficient of the cosine, which is 3. Amplitude represents half of the wave height, which is the distance from the mean level to either the crest or the trough.

Step 4: Calculate the Wave Height

The wave height is the vertical distance between the trough and the crest. Since the amplitude is 3, this accounts for the distance from the mean level to the crest and from the mean level to the trough, making the wave height \(2 \times 3 = 6\) feet.

Key Concepts

Period of a Function

The period of a function tells us how long it takes for the function to complete one full cycle. In the case of trigonometric functions like sine and cosine, the period is particularly important. It specifies how long it takes for the wave to repeat its pattern.

For the function \( h(t) = 3 \cos \left( \frac{\pi}{10} t \right) \), the formula for the period \( T \) is \( T = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( t \) in the argument of the cosine. Here, \( B = \frac{\pi}{10} \).

• To find the period, substitute \( B \) into the formula: \( T = \frac{2\pi}{\frac{\pi}{10}} \).
• This simplifies to \( T = \frac{2\pi}{1} \times \frac{10}{\pi} = 20 \).

Thus, the period of this cosine function is 20 seconds. This means every 20 seconds, the wave pattern repeats itself in the same manner.

Amplitude

The amplitude of a trigonometric function represents the height from the wave's mean position to its maximum (crest) or minimum (trough) levels. In the cosine function \( h(t) = 3 \cos \left(\frac{\pi}{10} t\right) \), the amplitude is given by the coefficient of the cosine function, which is 3. This means each crest or trough is 3 feet above or below the mean sea level.

• Amplitude is always a positive number.
• It is a crucial aspect in determining the wave's height.

The amplitude not only helps in understanding how tall the wave is but also in calculating the total vertical shift (wave height) between its crest and trough.

Wave Height

Wave height is a crucial concept in understanding wave motion. It is defined as the vertical distance from the bottom of the trough to the top of the crest of the wave. In other words, wave height is twice the amplitude. For the function \( h(t) = 3 \cos \left(\frac{\pi}{10} t\right) \), the amplitude is 3 feet.

• The wave height is calculated as \(2 \times \text{amplitude}\).
• In this case, it results in a wave height of \(2 \times 3 = 6\) feet.

Understanding wave height is essential because it gives a direct measurement of the size of the wave, important for marine navigation and safety.