As the moon orbits around us during the course of a month its distance from the Earth varies This lunar distance can be modeled as a sinusoidal function where X is time and f(x) Is the distance in thousands of kilometers. If: The moon's average distance from Earth is 385 thousand kilometers. The lunar distance varies by 20 thousand kilometers from average over the course of a month. At the beginning of this month, the moon's distance from Earth is increasing. It takes 7 days for the moon to reach its average distance from Earth. - The moon completes its orbit in 28 days. Determine the appropriate periodic function to model the distance of the moon from the Earth. Amplitude: Period: Phase Shift: Vertical Shift: square f(x)= square

To determine the appropriate periodic function to model the distance of the moon from the Earth, we need to consider the given information and use it to identify the amplitude, period, phase shift, and vertical shift of the sinusoidal function.<br /><br />Given information:<br />- The moon's average distance from Earth is 385 thousand kilometers.<br />- The lunar distance varies by 20 thousand kilometers from average over the course of a month.<br />- At the beginning of this month, the moon's distance from Earth is increasing.<br />- It takes 7 days for the moon to reach its average distance from Earth.<br />- The moon completes its orbit in 28 days.<br /><br />Step 1: Determine the amplitude.<br />The amplitude is the maximum deviation from the average distance, which is given as 20 thousand kilometers.<br /><br />Amplitude = 20 thousand kilometers<br /><br />Step 2: Determine the period.<br />The period is the time it takes for the moon to complete one orbit, which is given as 28 days.<br /><br />Period = 28 days<br /><br />Step 3: Determine the phase shift.<br />The phase shift is the horizontal shift of the sinusoidal function. Since the moon's distance is increasing at the beginning of the month, the phase shift will be positive.<br /><br />Phase Shift = 7 days<br /><br />Step 4: Determine the vertical shift.<br />The vertical shift is the average distance from Earth, which is given as 385 thousand kilometers.<br /><br />Vertical Shift = 385 thousand kilometers<br /><br />Now, we can write the appropriate periodic function to model the distance of the moon from the Earth:<br /><br />$f(x) = A \sin(B(x - C)) + D$<br /><br />Where:<br />- A is the amplitude<br />- B is the angular frequency<br />- C is the phase shift<br />- D is the vertical shift<br /><br />Substituting the values, we get:<br /><br />$f(x) = 20 \sin\left(\frac{2\pi}{28}(x - 7)\right) + 385$<br /><br />Therefore, the appropriate periodic function to model the distance of the moon from the Earth is:<br /><br />$f(x) = 20 \sin\left(\frac{2\pi}{28}(x - 7)\right) + 385$