Question

consider the graph above showing the percent of the moon visible at time \\(t\\), measured in days since an initial observation at day \\(t = 0\\) corresponding to the initial observation of a full moon on august 30th, 2023. suppose that the observation at point a occurred when \\(t = 14\\) and the observation at point c occurred when \\(t = 28\\).

approximately how many days occur between consecutive occurrences of 0% visibility, such as points a and e?

7 days
14 days
21 days
28 days

Explanation:

Identify the key points on the graph

Using the Periodic Functions and Mathematical Modeling knowledge points:

• At \(t = 0\), the visibility is at a maximum of \(100\%\) (full moon).
• Point A is the first minimum where visibility is \(0\%\) (new moon), occurring at \(t = 14\).
• Point C is the next maximum where visibility is \(100\%\), occurring at \(t = 28\).
• Point E is the second minimum where visibility is \(0\%\).

Determine the period of the function

Using the Periodic Functions knowledge point:

• The time from a maximum (\(t = 0\)) to the next consecutive maximum (Point C, \(t = 28\)) represents one full cycle (period, \(T\)).
• Therefore, the period is:

$$ T = 28 - 0 = 28\text{ days} $$

Calculate the time between consecutive minima

Using the Periodic Functions knowledge point:

• Consecutive occurrences of \(0\%\) visibility (such as points A and E) are consecutive minima of the periodic function.
• The time interval between any two consecutive minima is equal to one full period \(T\).
• Since \(T = 28\text{ days}\), the time between point A and point E is:

$$ t_E - t_A = T = 28\text{ days} $$

Answer:

• 7 days
• 14 days
• 21 days
• 28 days (Correct answer)