Question

1. the hours of daylight in glasgow, scotland, can be modeled by the function ( f ) given by the table where ( d ) is the number of days after june 21, 2017, and ( f(d) ) is the number of hours on that day.

a. what is the amplitude of ( f )? what does it mean in this context? (6 pts)

b. what is the period of ( f )? what does it mean in this context? (6 pts)

c. what is the midline of ( f )? what does it mean in this context? (6 pts)

the table is as follows:

( d )	0	91	183	274	365	456	548	639	730	821	913	1004	1095

Explanation:

Part a: Amplitude of \( f \)

Step 1: Identify Max and Min Values

From the table, the maximum value of \( f(d) \) is \( 17.58 \) and the minimum value is \( 6.98 \).

Step 2: Calculate Amplitude

The formula for amplitude \( A \) of a periodic function is \( A=\frac{\text{Max}-\text{Min}}{2} \). Substituting the values: \( A = \frac{17.58 - 6.98}{2}=\frac{10.6}{2} = 5.3 \).

In context, the amplitude represents the maximum deviation of the number of daylight hours from the midline (average) over a year. It shows how much the daylight hours can vary above or below the average.

Step 1: Observe the Pattern

Looking at the table, the values of \( f(d) \) repeat. For example, at \( d = 0 \), \( f(d)=17.58 \); at \( d = 365 \), \( f(d)=17.58 \); at \( d = 730 \), \( f(d)=17.58 \), etc. Similarly, other values repeat at intervals of \( 365 \) days.

Step 2: Determine Period

The period is the length of one full cycle. Here, the function repeats every \( 365 \) days. In context, this means the pattern of daylight hours in Glasgow repeats every \( 365 \) days (a year), which is consistent with the annual cycle of seasons.

Step 1: Calculate Midline

The midline (vertical shift) \( M \) of a periodic function is the average of the maximum and minimum values. Using \( \text{Max}=17.58 \) and \( \text{Min}=6.98 \), we have \( M=\frac{17.58 + 6.98}{2}=\frac{24.56}{2}=12.28 \).

Step 2: Interpret in Context

In this context, the midline represents the average number of daylight hours in Glasgow over a year. It is the central value around which the daylight hours oscillate (increase and decrease) throughout the year.

Answer:

Amplitude is \( 5.3 \). It represents the maximum deviation of daylight hours from the average.

Part b: Period of \( f \)