The moon's illumination changes in a periodic way that can be modeled by a trigonometric function. On the night of a full moon, the moon provides about 0.25 lux of illumination (lux is the SI unit of illuminance). During a new moon, the moon provides 0 lux of illumination. The period of the lunar cycle is 29.53 days long. The moon will be full on December 25,2015. Note that December 25 is 7 days before January 1. Find the formula of the trigonometric function that models the illumination L of the moon t days after January 1,2016 . Define the function using radians. L(t)=◻

Question

The moon's illumination changes in a periodic way that can be modeled by a trigonometric function. On the night of a full moon, the moon provides about 0.25 lux of illumination (lux is the SI unit of illuminance). During a new moon, the moon provides 0 lux of illumination. The period of the lunar cycle is 29.53 days long. The moon will be full on December  25,2015. Note that December 25 is 7 days before January 1. Find the formula of the trigonometric function that models the illumination  L of the moon  t days after January 1,2016 . Define the function using radians.  L(t)=◻

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Determine Amplitude: Determine the amplitude of the trigonometric function. The amplitude (A) is half the difference between the maximum and minimum values of the function. Since the maximum illumination is 0.25 lux (full moon) and the minimum is 0 lux (new moon), the amplitude is: A = (0.25 - 0) / 2 = 0.25 / 2 = 0.125 lux.Determine Period: Determine the period of the trigonometric function. The period (T) of the lunar cycle is given as 29.53 days. In terms of radians, the period of a sine or cosine function is 2π, so we need to find the value that will stretch or compress the function to fit the lunar cycle. The period in the function will be: T = 2π / B, where B is the horizontal stretch factor. To find B, we rearrange the equation: B = 2π / T. B = 2π / 29.53.Calculate Stretch Factor: Calculate the horizontal stretch factor (B). B = 2π / 29.53 ≈ 0.2124 radians per day.Determine Shift: Determine the horizontal shift (phase shift) of the function. Since the full moon is on December 25, 2015, and we are considering t days after January 1, 2016, the phase shift (D) will be the number of days from the full moon to January 1. This is 7 days since December 25 is 7 days before January 1. The function will be at its maximum 7 days before t = 0, so the phase shift is -7 days.Determine Vertical Shift: Determine the vertical shift of the function. The vertical shift (C) is the average of the maximum and minimum values of the function. Since the maximum is 0.25 lux and the minimum is 0 lux, the vertical shift is: C = (0.25 + 0) / 2 = 0.25 / 2 = 0.125 lux.Write Function: Write the trigonometric function. We can use a cosine function because it starts at its maximum value, which corresponds to the full moon. The general form of the cosine function is: L(t) = A * cos(B(t - D)) + C. Substituting the values we have: A = 0.125 lux, B ≈ 0.2124 radians per day, D = -7 days, C = 0.125 lux. L(t) = 0.125 * cos(0.2124(t + 7)) + 0.125.

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