Problem 59 - 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 3.00 P.M.? (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The height of the tide at 3:00 P.M. is 7.4 feet.

Step 1: Identify the Variable for Calculation

The problem requires calculating the height of the tide at 3:00 P.M. We first determine the value of the variable \(x\). Since \(x\) represents the hours since midnight, 3:00 P.M. corresponds to 15 hours after midnight. Therefore, \(x = 15\).

Step 2: Substitute the Hour Into the Formula

We substitute \(x = 15\) into the given height function: \[h(15) = 5 + 4.8 \sin \left[\frac{\pi}{6}(15 + 4)\right].\]

Step 3: Simplify the Expression Inside the Sine Function

Calculate the expression inside the sine function:\[\frac{\pi}{6}(15 + 4) = \frac{\pi}{6} \times 19 = \frac{19\pi}{6}.\]

Step 4: Calculate the Sine Value

Compute \(\sin\left(\frac{19\pi}{6}\right)\). Note that since \(\frac{19\pi}{6}\) is equivalent to \(\frac{\pi}{6}\) (since \(\frac{19\pi}{6} - 3\pi = \frac{\pi}{6}\)), we have:\[\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}.\]

Step 5: Calculate the Height of the Tide

Substitute the sine value back into the equation:\[h(15) = 5 + 4.8 \times \frac{1}{2} = 5 + 2.4 = 7.4.\] Therefore, the height of the tide at 3:00 P.M. is 7.4 feet.

Key Concepts

Tides and Water Levels

Tides are the rhythmic rise and fall of sea levels caused by the gravitational pull of the moon and the sun, as well as the rotation of the Earth. In harbors and coastal areas, tides follow a predictable cyclical pattern, which impacts shipping and beach activities. Water levels can vary significantly depending on the time of day and the tide cycle.

Understanding how tides work is important because the height of the water affects many maritime activities. For instance, tides can determine the docking and departure times of ships and influence when it's safe to go swimming or fishing in the ocean.

• The highest point reached by a tide is called high tide.
• The lowest point is known as low tide.
• Tides usually occur in a semi-diurnal pattern, featuring two high and two low tides each day.

By knowing when these tides happen and how high or low they can be, we can plan activities around them. This knowledge is crucial for harbor management and coastal engineering, ensuring we can safely navigate and enjoy our coastal environments.

Sine Function

The sine function is a mathematical function that originates from trigonometry. It describes a smooth, periodic oscillation. This function is fundamental in modeling various cyclical phenomena, like tides, sound waves, and electrical currents.

The sine function is usually written as \( \sin(x) \), where \( x \) is the angle measured in radians. Its graph is a wave that repeats every \( 2\pi \) radians. The sine function oscillates between -1 and 1, making it perfect for modeling periodic processes.

• The peak represents the maximum value of the function.
• The trough is the minimum value.
• The midline is the average or central level the wave oscillates around.

The equation for the height of water in the given exercise, \( h(x) = 5 + 4.8 \sin \left[ \frac{\pi}{6}(x+4) \right] \), is a variation of the basic sine function, adjusted to reflect the pattern of tides over time, with specific changes in amplitude and phase shift.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world phenomena. It allows us to predict and analyze behaviors without physically observing them at every moment, saving time and resources. In the context of tides, mathematical models help us understand how tides will behave in the future, which is crucial for planning maritime activities.

In our exercise, the function \( h(x) = 5 + 4.8 \sin \left[ \frac{\pi}{6}(x+4) \right] \), models the height of tides over time. It involves:

• Amplitude: This represents the range of height changes. In this function, the water level varies by 4.8 feet.
• Phase Shift: The term \( x+4 \) in the sine function denotes a horizontal shift, adjusting the timing of the high and low tides.
• Vertical Shift: The +5 adjusts the base level, setting the mean tide height at 5 feet.

By modeling tides with this formula, we gain insightful predictions about water levels throughout the day, aiding in strategic planning for maritime operations.

Trigonometry in Real World Applications

Trigonometry is not just about solving triangles or theoretical computations; it has profound applications in the real world. It plays a crucial role in understanding natural phenomena and enhancing technological capabilities.

Examples of real-world applications include:

• Navigation: Pilots and sailors use trigonometric functions to chart their courses.
• Architecture: Trigonometry helps design buildings, focusing on structural integrity and aesthetic appeal.
• Astronomy: Trigonometric concepts calculate distances to stars and celestial movements.
• Engineering: It aids in constructing everything from bridges to rockets.

In our specific case with tides, trigonometry offers the ability to forecast water levels through models, ensuring safer harbors and well-timed maritime activities. Thus, trigonometry is not just a mathematical curiosity; it's a vital tool for solving practical, everyday problems.