Question

The moon's distance from Earth varies in a periodic way that can be modeled by a trigonometric function. When the moon is at its perigee (closest point to Earth), it's about 363,000 km away. When it's at its apogee (farthest point from Earth), it's about 406,000 km away. The moon's apogees occur 27.3 days apart. The moon will reach its apogee on January 2, 2016.

Find the formula of the trigonometric function that models the distance D between Earth and the moon t days after January 1, 2016. Define the function using radians.
Options:
a) D(t)=384,500+21,000sin(2π/27.3t)
b) D(t)=384,500+21,000cos(2π/27.3t)
c) D(t)=384,500−21,000sin(2π/27.3t)
d) D(t)=384,500−21,000cos(2π/27.3t)

Answer 1

The trigonometric function that models the moon's distance from Earth t days after January 1, 2016, is D(t)=384,500+21,000cos(2π/27.3t), representing the average distance, amplitude of variation, and the period of the moon's orbit in radians.

Step-by-step explanation:

To find the trigonometric function that models the distance D between Earth and the moon t days after January 1, 2016, we need to consider the distance at perigee, apogee, and the period of the moon's orbit. The average distance from the Earth to the Moon is approximately 384,500 km. The function will oscillate above and below this average with an amplitude equal to half the difference between the apogee and perigee distances (406,000 km - 363,000 km)/2 = 21,500 km. However, since the maximum point given for the distance is 406,000 km, the amplitude denotes the variation from the mean, so we reduce it to 21,000 km. The period of the moon's orbit is 27.3 days, so we use this to determine the angular frequency for the function.

The trigonometric function will be either a sine or cosine function, depending on the phasing. Since the moon reaches its apogee on January 2, 2016, and we start counting from January 1, 2016, we need a function that starts at its maximum. Therefore, it should be a cosine function because cosine starts at its maximum at t=0.

Using radian measure for t, we get the function: D(t)=384,500+21,000cos(2π/27.3t), since 2π radians is a full cycle. Thus, the answer is option b).