Math Problem Statement

Write an equation in terms of y, average daylight in minutes, and t, time in months, to represent the given data in context.

Solution

To create an equation for the average daylight yy in terms of tt (in months), we need to analyze the periodic nature of the data shown in the graph.

Observations:

  1. Amplitude: The maximum daylight is approximately 835 minutes, and the minimum is around 497 minutes. The amplitude AA of this function is half the difference between these values. A=8354972=169A = \frac{835 - 497}{2} = 169

  2. Midline (Vertical Shift): The midline of the function, which is the average of the maximum and minimum values, is given by: D=835+4972=666D = \frac{835 + 497}{2} = 666

  3. Period: The period of the function is 12 months since the pattern repeats every year. In a sine or cosine function, the period TT is given by T=2πBT = \frac{2\pi}{B}, where BB is the frequency. Here, T=12T = 12, so: B=2π12=π6B = \frac{2\pi}{12} = \frac{\pi}{6}

  4. Phase Shift: Observing the graph, a maximum occurs at t=11.5t = 11.5 months. For a cosine function, which typically starts at a maximum, we can use a phase shift of 11.511.5 months.

Equation

With these parameters, we can write the equation as a cosine function: y=Acos(B(tC))+Dy = A \cos(B(t - C)) + D Substituting the values: y=169cos(π6(t11.5))+666y = 169 \cos\left(\frac{\pi}{6}(t - 11.5)\right) + 666

Final Answer

The equation for the average daylight yy in terms of tt is: y=169cos(π6(t11.5))+666y = 169 \cos\left(\frac{\pi}{6}(t - 11.5)\right) + 666

This equation models the average daylight in minutes over time tt in months.

Would you like further details or have any questions about this solution?


Here are five related questions for further exploration:

  1. How would the equation change if the period of daylight varied across years?
  2. What would be the effect on the graph if the phase shift was adjusted?
  3. Can this cosine model be used for different cities, or would we need different parameters?
  4. How would you determine the equation if you only knew the midline and amplitude?
  5. 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