Part 1: Modeling Sunrise (6 points)
Given Function:
\[ y = 12 + 1.5 \sin(2365x - 2) \]
Step 1: Identify Characteristics of Sinusoidal Function
- Amplitude: 1.5 hours (the maximum deviation from the midline).
- Vertical Shift: 12 hours (the average time of sunrise).
- Period: Calculate using the formula \( \text{Period} = \frac{2\pi}{k} \) where \( k = 2365 \). The period is thus: \[ \text{Period} = \frac{2\pi}{2365} \approx 0.00265 \text{ years} \]
Step 2: Graphing the Function Using a graphing program, we graph the function over one year (x from 0 to 1):
- X-axis: Days of the year (0 to 365).
- Y-axis: Sunrise time (y in hours, from 0 to 20).
- Graph: (You can create a graph of this function online or using a graphing calculator.)
Step 3: Earliest and Latest Sunrise Times
- Maximum and Minimum Values:
- Maximum \( y = 12 + 1.5 = 13.5 \) hours at \( \sin(2365x - 2) = 1 \)
- Minimum \( y = 12 - 1.5 = 10.5 \) hours at \( \sin(2365x - 2) = -1 \)
Determining when these occur:
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Maximum occurs when: \[ 2365x - 2 = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \] Solving for max yields specific days.
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Minimum occurs when: \[ 2365x - 2 = \frac{3\pi}{2} + 2n\pi \]
By solving these equations for \( x \), we get the days of the year corresponding to the earliest and latest sunrise.
Step 4: Model Explanation This sine function gives a clear model of the gradual changes in sunrise time through the year, reflective of the Earth's tilt and orbit.
Step 5: Changing the Model
- Increase the amplitude: Alter the function to \( y = 12 + A \sin(2365x - 2) \) where \( A > 1.5 \).
- Change the vertical shift: Adjust the midline using a higher value than 12 to simulate later sunrises.
Part 2: Wave Behavior (6 points)
Given Wave Functions:
\[ y_1 = 2\sin(x) \]
\[ y_2 = 2\sin(x) + 2 \]
Step 1: Graphing the Waves
- Graph: Use graphing software to plot both wave equations and their resultant \( y_R = y_1 + y_2 \).
- Label each wave clearly.
Step 2: Identifying Interference Patterns
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Constructive Interference occurs when the peaks of both waves align:
- This happens when both \( \sin(x) \) values are maximally aligned, which can be calculated by finding peaks where both functions output the same maximum.
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Destructive Interference occurs at points where the waves cancel:
- This happens when the calculated values of \( y_1 \) and \( y_2 \) result in zero at certain points, primarily where \( 2\sin(x) + 2 - 2\sin(x) = 0 \).
Part 3: Resonance (6 points)
Given Waves:
\[ y_1 = 3\sin(2x) \]
\[ y_2 = 3\sin(2x + 3) \]
Step 1: Resultant Wave Using trigonometric identities, we can deduce: \[ y_R = 3\sin(2x) + 3\sin(2x + 3) \] (Apply sum-to-product to find the simplified form).
Step 2: Maximum Amplitude The resultant maximum amplitude will be determined by the coefficients of the sinusoidal components, which can total to \( 3 \times 2 = 6 \) for maximal constructive interference.
Step 3: Calculate Maximum Occurrence To find when \( y_R \) is at maximum: Set \( 2x + C = \frac{\pi}{2} + 2n\pi \). By solving for \( x \):
- Substitute \( C = 3 \) from the second wave.
Reflection (2 points)
In summary, the applications of trigonometric functions such as sine and cosine are pivotal in modeling natural phenomena like sunrise times and wave behaviors. This understanding allows for the prediction of various real-world events across numerous fields, including sound engineering (as in beats and noise cancellation) and environmental science (modeling natural oscillations). These functions provide a valuable framework for analyzing periodic behavior, which is essential for creating systems that can predict and respond to patterns in nature or technology.