Problem 46 - Algebra and Trigonometry

Question

The height of the water, \(H,\) in feet, at a boat dock \(t\) hours after \(6 \mathrm{A}\). M. is given by $$ H=10+4 \sin \frac{\pi}{6} t $$ a. Find the height of the water at the dock at \(6 \mathrm{A} . \mathrm{M} ., 9 \mathrm{A.M}\). noon, 6 P.M., midnight, and 3 A.M. b. When is low tide and when is high tide? c. What is the period of this function and what does this mean about the tides?

Step-by-Step Solution

Answer

a. The water heights at 6 A.M., 9 A.M., noon, 6 P.M., midnight, 3 A.M. are calculated by substituting \(t\) in the equation. b. The high tide occurs when \(\sin \frac{\pi}{6} t =1\) and low tide when \(\sin \frac{\pi}{6} t =-1\). Our calculation will determine the times when these occur. c. The period of this function is 12 hours, which indicates that the tide cycle (from high tide to low tide and back to high tide) repeats every 12 hours.

Step 1: Determine the height at the specific time points

We can substitute the different \(t\) values into the equation \(H=10+4 \sin \frac{\pi}{6} t\) to find the water heights. At 6 A.M., 9 A.M., noon, 6 P.M., midnight, and 3 A.M., \(t\) is 0, 3, 6, 12, 18, and 21 respectively.

Step 2: Identify the timing of high tide and low tide

Water height is at an extreme when the sine function is at its maximum and minimum, which are 1 and -1 respectively. We can equate \(\sin \frac{\pi}{6} t\) to these values and solve for \(t\).

Step 3: Find period of the function and interpret it

The period of a sine function of the form \(A \sin Bt + C\) is given by \(\frac{2\pi}{|B|}\). Thus, the period of our function would be found by \(\frac{2\pi}{\frac{\pi}{6}} = 12\). This represents how frequently the tide cycle repeats.

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions and is often written as \( \sin(x) \). It is a periodic function, which means that it repeats its pattern over regular intervals.
In the context of our problem, the sine function is part of the equation for tide heights: \[ H = 10 + 4 \sin\left( \frac{\pi}{6} t \right) \]Here, the sine function shows how the height of the tide changes through time \(t\).
The function oscillates smoothly between +1 and -1 values.

• This means that the value of \(4\sin\left(\frac{\pi}{6} t \right)\) varies between -4 and 4.
• Adding 10 shifts this upwards, so the tide height \(H\) ranges from 6 to 14 feet.

The sine function helps model natural phenomena like tides due to its cyclical nature, capturing how tides rise to a peak and fall to a low repeatedly.

Period of a Function

The period of a function is the length of one complete cycle of the function before it starts repeating itself. It's a crucial concept when understanding the sine function and the tide model given in our problem.
For sine functions like \( A \sin(Bt + C) \), the period \( P \) is calculated as:\[ P = \frac{2\pi}{|B|} \]In our tide model, \( B = \frac{\pi}{6} \), so the period is:\[ P = \frac{2\pi}{\frac{\pi}{6}} = 12 \]This indicates that the cycle of tides - one high and one low - repeats every 12 hours.

• This regular repetition is why you can predict when the next high or low tide will occur after observing one.
• Understanding the period helps in planning activities around tidal variations, such as docking boats or coastal fishing.

Tides and Water Heights

Tides are caused by the gravitational pull of the moon and the sun on the earth's oceans. They cause regular rising and lowering of water levels, which can be predicted using trigonometric functions like sine.
In our equation, \( H = 10 + 4 \sin\left( \frac{\pi}{6} t \right) \), the water height is modeled over time:

• The term \( 10 \) is the average water height.
• The sine component \( 4 \sin\left( \frac{\pi}{6} t \right) \) models the fluctuation above and below this average.

This model helps to predict the precise water height at various times:
At the start of the cycle (6 A.M.), the tide is average, while subsequent calculations show how it varies through the day.

• You can calculate exact water heights by substituting specific \( t \) values into the equation.
• This predictive capability is critical for maritime activities and understanding coastal environments.

Extreme Values of Trigonometric Functions

Trigonometric functions like sine reach extreme values at certain points in their cycles. In the sine function, these extremes are +1 and -1.
In the context of our tide problem, these extreme values dictate the tide conditions:

• At \( \sin \left( \frac{\pi}{6} t \right) = 1 \), the tide is at its highest point.
• Conversely, when \( \sin \left( \frac{\pi}{6} t \right) = -1 \), the tide is at its lowest.

By substituting these extreme sine values into the equation, the water levels are maximized and minimized:

• High Tide: \( H = 10 + 4(1) = 14 \) feet.
• Low Tide: \( H = 10 + 4(-1) = 6 \) feet.

Recognizing these extremes is vital for predicting tidal conditions at docks, ensuring safe navigation, and avoiding hazards.