Problem 224 - Calculus Volume 1

Question

[T] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function $$ D(t)=5 \sin \left(\frac{\pi}{6} t-\frac{7 \pi}{6}\right)+8 $$ where \(t\) is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 \(\mathrm{ft}\) .

Step-by-Step Solution

Answer

The first time after midnight when the depth is 11.75 ft is at 9 a.m.

Step 1: Set Up the Equation

To find the first time after midnight when the depth is 11.75 ft, we need to set the equation for the depth equal to 11.75 ft. The equation is: \[ 5 \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 8 = 11.75 \] Subtract 8 from both sides.

Step 2: Simplify the Equation

After subtracting 8 from both sides, the equation becomes: \[ 5 \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) = 3.75 \] Now, divide both sides by 5 to isolate the sine function.

Step 3: Isolate the Sine Function

Divide by 5, resulting in: \[ \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) = 0.75 \] Now find the angle whose sine is 0.75.

Step 4: Solve for the Angle

Using the inverse sine function, we find: \[ \frac{\pi}{6} t - \frac{7\pi}{6} = \sin^{-1}(0.75) \] The principal value of \( \sin^{-1}(0.75) \) is \( \frac{\pi}{3} \). Therefore: \[ \frac{\pi}{6} t - \frac{7\pi}{6} = \frac{\pi}{3} \]

Step 5: Solve for t

Add \( \frac{7\pi}{6} \) to both sides to solve for \( \frac{\pi}{6} t \): \[ \frac{\pi}{6} t = \frac{\pi}{3} + \frac{7\pi}{6} = \frac{2\pi}{6} + \frac{7\pi}{6} = \frac{9\pi}{6} = \frac{3\pi}{2} \]Multiply both sides by \( \frac{6}{\pi} \) to solve for \( t \): \[ t = 3 \times 2 = 9 \]

Step 6: Verify the Time

Check the calculation by plugging \( t = 9 \) into the original function to confirm: \[ D(9) = 5 \sin \left( \frac{\pi}{6} \times 9 - \frac{7\pi}{6} \right) + 8 = 11.75 \] Substitute and calculate: \[ 5 \sin \left(\frac{3\pi}{2} - \frac{7\pi}{6}\right) + 8 = 11.75 \] This checks out, confirming that \( t = 9 \) gives a depth of 11.75 ft.

Key Concepts

Tide Modeling

Tide modeling is a fascinating application of trigonometric functions used to understand and predict the repeating rise and fall of sea levels. This concept is based on the fact that tides are cyclical, occurring with regular intervals and variations caused by the gravitational pull of the moon and sun. By using mathematical functions, we can model these patterns accurately. In our context, the function that describes the tide at a dock is given by
\[ D(t) = 5 \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 8 \]This equation allows us to find the water depth for any given time, where:

• \( 5 \): Amplitude of the sine function, indicating the maximum variation in water depth.
• \( \frac{\pi}{6} \): Represents the tidal period in relation to time \( t \), showing how frequently the tide cycles.
• \( -\frac{7\pi}{6} \): Phase shift, which adjusts the starting point of the tide cycle relative to midnight.
• \( +8 \): Vertical shift, which changes the central axis of the sine wave.

Using tide modeling, we can predict the exact time and level of tides, which is essential for activities such as docking boats and coastal planning.

Sine Function

The sine function is one of the foundational aspects of trigonometry and plays a significant role in modeling periodic phenomena like tides. The function \( \sin(x) \) maps angles to their respective sine values, depicting a smooth and continuous wave.
In the equation used in tide modeling, \( D(t) = 5 \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 8 \), the sine function creates the wave pattern observed in tides:

• Amplitude: The coefficient \( 5 \) indicates the wave's tallest peak and deepest trough.
• Period: Defined by the term \( \frac{\pi}{6} \), specifying how long it takes for the sine wave to complete one cycle. The smaller this value, the quicker the wave repeats.
• Phase Shift: The term \( -\frac{7\pi}{6} \) shifts the wave horizontally along the time axis, determining when the cycle begins.

As a fundamental trigonometric function, \( \sin(x) \) is instrumental in representing and solving a wide array of wave-related problems in physics and engineering.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for determining angles when you know the value of a trigonometric function. They allow us to "reverse" the operations of standard trig functions. In our tide modeling case, we use the inverse sine function to isolate the angle that produces a known sine value.
For example, once simplified, our problem required finding the time \( t \) when the sine function value was 0.75:\[ \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) = 0.75 \]By utilizing the inverse function, \( \sin^{-1}(0.75) \), we can find the angle:

• The principal value here is \( \frac{\pi}{3} \), giving us a specific angle for the sine equation.
• This angle helps us determine the values of \( t \) that result in a depth of 11.75 feet in the dock scenario.

Inverse trigonometric functions, therefore, provide a powerful way to transition from function values back to angles, enabling solutions to problems where angles are not immediately known.