Problem 60 - Precalculus

Question

Refer to the following: The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula \(h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]\) where \(x\) is the number of hours since midnight and \(h\) is the height of the tide in feet. What is the height of the tide at 5.00 A.M.?

Step-by-Step Solution

Answer

The height of the tide at 5:00 A.M. is 0.2 feet.

Step 1: Identify the value of x

First, we need to determine the value of \(x\) for 5:00 A.M. Since \(x\) represents the number of hours since midnight, 5:00 A.M. corresponds to \(x = 5\).

Step 2: Substitute x into the formula

Now, substitute \(x = 5\) into the function for the height of the tide: \[ h(5) = 5 + 4.8 \sin \left[ \frac{\pi}{6}(5+4) \right] \].

Step 3: Simplify the expression within the sine

Calculate the expression inside the sine function: \(5 + 4 = 9\), then \(\frac{\pi}{6} \times 9 = \frac{9\pi}{6} = \frac{3\pi}{2}\).

Step 4: Evaluate the sine function

Find the sine of \(\frac{3\pi}{2}\). This angle corresponds to 270° on the unit circle, where the sine value is \(-1\).

Step 5: Complete the formula substitution

Substitute \(-1\) into the height formula: \[ h(5) = 5 + 4.8 (-1) \].

Step 6: Calculate the height of the tide

Perform the arithmetic to find \( h(5) = 5 - 4.8 = 0.2 \) feet.

Key Concepts

Sine Function

The sine function is an essential component of trigonometry, widely used to model periodic phenomena such as sound waves or tidal movements.

• The sine function is denoted as \( \sin(\theta) \), where \( \theta \) is an angle that can be measured in degrees or radians.
• It is a periodic function with a period of \( 2\pi \) radians (or 360°), meaning its values repeat every \( 2\pi \) radians.

The sine function varies between -1 and 1. At angles like \( \frac{\pi}{2} \) or 90°, the sine value reaches its maximum, which is 1. Conversely, at angles like \( \frac{3\pi}{2} \) or 270°, the sine value hits its minimum, which is -1. This property is key in many applications, including the tide height formula where it adjusts the base height according to periodic changes.

Tide Height Calculation

Calculating the height of the tide using a trigonometric function involves understanding how the sine wave is applied in the context of time.

• The formula provided was: \[ h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right] \]
• This represents a combination of amplitude scaling and vertical shifts based on the sine wave.

Here’s how the function breaks down:- **Base Height:** The number 5 represents the average base height of the tide when there are no additional effects from the sine component.- **Amplitude:** The 4.8 before the sine function scales the wave, determining how much above or below the base height the tide can reach. - **Phase Shift:** The addition of 4 inside the sine function shifts the function horizontally, affecting the timing of the wave’s peaks and troughs.- **Frequency:** The \( \frac{\pi}{6} \) affects how quickly the wave repeats.For example, when finding the tide height at 5 A.M. (or \( x = 5 \)), substituting into the formula, you first simplify within to find \( \frac{3\pi}{2} \), where the sine itself evaluates to -1. Thus, the calculated tide height is close to the low point on the curve, resulting in a height of 0.2 feet.

Unit Circle

The unit circle is a fundamental concept for understanding trigonometric functions, especially useful for visualizing the sine function in this context.

• A unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.
• The angle \( \theta \) is measured from the positive x-axis, counter-clockwise around the circle.

In the unit circle:- The x-coordinate of a point on the circle represents \( \cos(\theta) \).- The y-coordinate represents \( \sin(\theta) \).When we look at specific angles like \( \frac{3\pi}{2} \), this corresponds to the point where the circle touches the negative y-axis, making the sine value -1. Understanding the unit circle allows for easy conversion between angles and their corresponding sine and cosine values. Thus, it helps in solving trigonometric equations and visualizing these functions, such as when calculating tide heights.