Math Problem Statement
Solution
To create an equation for the average daylight in terms of (in months), we need to analyze the periodic nature of the data shown in the graph.
Observations:
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Amplitude: The maximum daylight is approximately 835 minutes, and the minimum is around 497 minutes. The amplitude of this function is half the difference between these values.
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Midline (Vertical Shift): The midline of the function, which is the average of the maximum and minimum values, is given by:
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Period: The period of the function is 12 months since the pattern repeats every year. In a sine or cosine function, the period is given by , where is the frequency. Here, , so:
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Phase Shift: Observing the graph, a maximum occurs at months. For a cosine function, which typically starts at a maximum, we can use a phase shift of months.
Equation
With these parameters, we can write the equation as a cosine function: Substituting the values:
Final Answer
The equation for the average daylight in terms of is:
This equation models the average daylight in minutes over time in months.
Would you like further details or have any questions about this solution?
Here are five related questions for further exploration:
- How would the equation change if the period of daylight varied across years?
- What would be the effect on the graph if the phase shift was adjusted?
- Can this cosine model be used for different cities, or would we need different parameters?
- How would you determine the equation if you only knew the midline and amplitude?
- What is the significance of using cosine instead of sine in this model?
Tip: When analyzing periodic functions, identifying the amplitude, period, and phase shift can help you quickly construct an accurate model.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Periodic Functions
Amplitude
Vertical Shift
Phase Shift
Formulas
Amplitude (A) = (Max - Min) / 2
Midline (Vertical Shift) (D) = (Max + Min) / 2
Period (T) = 2π / B
Phase Shift (C) based on cosine function starting point
Cosine equation: y = A * cos(B(t - C)) + D
Theorems
Properties of Cosine Function
Suitable Grade Level
Grades 10-12
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