Problem 224 - Calculus Volume 1

Question

The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function \(D(t)=5 \sin \left(\frac{\pi}{6} t-\frac{7 \pi}{6}\right)+8, \quad\) 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 9 hours.

Step 1: Set the Equation to the Given Depth

We need to find the first time after midnight when the depth is 11.75 ft. Set the function equal to 11.75: \[5 \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 8 = 11.75\] Then, solve for the sine component by subtracting 8 from both sides: \[5 \sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) = 3.75\] Divide both sides by 5 to isolate the sine function: \[\sin \left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) = 0.75\]

Step 2: Solve for the Sine Argument

Next, we find the angle that gives us the sine value of 0.75. Recall that \(\sin^{-1}(0.75)\) gives us the reference angle. Let's solve for this: \[\frac{\pi}{6} t - \frac{7\pi}{6} = \sin^{-1}(0.75)\] The principal value (first quadrant) for \(\sin^{-1}(0.75)\) is approximately \(\frac{\pi}{3}\). Thus, we have: \[\frac{\pi}{6} t - \frac{7\pi}{6} = \frac{\pi}{3}\]

Step 3: Solve for t in the Principal Value Case

Add \(\frac{7\pi}{6}\) to both sides to solve for \(t\): \[\frac{\pi}{6} t = \frac{\pi}{3} + \frac{7\pi}{6}\] Convert \(\frac{\pi}{3}\) to a common denominator to combine the fractions: \[\frac{\pi}{3} = \frac{2\pi}{6}\] Thus: \[\frac{\pi}{6} t = \frac{2\pi}{6} + \frac{7\pi}{6} = \frac{9\pi}{6}\] Divide through by \(\frac{\pi}{6}\) to isolate \(t\): \[t = 9\]

Step 4: Check for Additional Solutions

Since sine is periodic with period \(2\pi\), additional solutions can be found by using \[\frac{\pi}{6} t - \frac{7\pi}{6} = \frac{\pi}{3} + 2n\pi\] where \(n\) is an integer. Solve for \(t\): \[t = 9 + 12n\] For \(n = 0\), we get \(t = 9\), which is the first time after midnight. For \(n = 1\), \(t = 21\), which is past \(t = 9\). Therefore, the first time is at \(t = 9\).

Key Concepts

Periodic Functions

Periodic functions are fascinating mathematical concepts that repeat their values at regular intervals. Think of them as a way to model cyclical patterns you see in daily life, such as the rising and setting of the sun, tides, or the seasons. In this exercise, the function that models the change in water depth at a dock is periodic.

• Periodic functions have a period, which is the smallest positive interval after which the function repeats.
• The period is linked to the concept of frequency, which describes how often a value repeats in a given time frame.
• Common examples of periodic functions include sine and cosine functions.

The function given in the exercise, \( D(t) = 5\sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 8 \), is periodic because it involves the sine function, which is known for its wave-like pattern. In this scenario, the periodic nature of the sine function helps predict the water depth at any time "\( t \)" after midnight. This repeating pattern allows us to pinpoint the times when certain water depths, like 11.75 feet, occur.

Sine Function

The sine function is one of the key trigonometric functions. It plays a critical role in modeling waves and oscillations. In this exercise, the sine function is used to model tidal changes, providing a realistic representation of how tides rise and fall throughout the day.

• The sine function is periodic with a period of \(2\pi\), repeating its values every \(360^\circ\) or \(2\pi\) radians.
• The function oscillates smoothly between -1 and 1, making it ideal for modeling natural, smooth wave patterns.

You encounter the sine function in the given equation as part of \( 5\sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) \). This transforms into a sine wave that oscillates above and below a centerline of "8", due to the constant added to the sine function, shifting the average value higher. This sine function characterizes how drastically the tides change over time and helps identify when specific tidal levels, like 11.75 feet, occur.

Inverse Trigonometric Functions

Inverse trigonometric functions enable us to work backwards from the value of a trigonometric function to find the angle that produced it. In simpler terms, if you know the sine of an angle, you can use the inverse sine (\(\sin^{-1}\)) to find that angle.

• The principal value of the inverse sine function, \(\sin^{-1}(x)\), is limited to \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• This helps in determining the specific angle within that range whose sine is \(x\).

For the given problem, to find when the water depth reaches 11.75 feet, it's solved using \(\sin^{-1}(0.75)\), derived from the equation \(\sin\left(\frac{\pi}{6}t - \frac{7\pi}{6}\right) = 0.75\). The inverse sine function helps determine the reference angle that leads to this tidal value. Using these concepts, we can pinpoint the exact time after midnight when the desired depth occurs.