Hours of daylight in Fairbanks If the formula for D(t) in Example 12 is used for Fairbanks...

Hours of daylight in Fairbanks If the formula for D(t) in Example 12 is used for Fairbanks, Alaska, then K ≈ 12. Sketch the graph of D in this case for 0 ≤ t ≤ 365.

EXAMPLE 12 Approximating the number of hours of daylight in a day

The number of hours of daylight D(t) at a particular time of the year can be approximated by

for t in days and t = 0 corresponding to January 1. The constant K determines the total variation in day length and depends on the latitude of the locale.

(a) For Boston, K ≈6. Sketch the graph of D for 0 ≤ t ≤365.

(b) When is the day length the longest? the shortest?

SOLUTION

(a) If K = 6, then K/2 = 3, and we may write D(t) in the form

where

We shall sketch the graph of f and then apply a vertical shift through a distance 12.

As in part (2) of the theorem on amplitudes, periods, and phase shifts, we can obtain a t-interval containing exactly one cycle by solving the following inequality:

Hence, one sine wave occurs on the interval [79, 444]. Dividing this interval into four equal parts, we obtain the following table of values, which indicates the familiar sine wave pattern of amplitude 3.

If t = 0,

Since the period of f is 365, this implies that f(365) ≈−2.9.

The graph of f for the interval [0, 444] is sketched in Figure 13, with different scales on the axes and t rounded off to the nearest day.

Applying a vertical shift of 12 units gives us the graph of D for 0 ≤ t ≤ 365 shown in Figure 13.

FIGURE 13

(b) The longest day—that is, the largest value of D(t)—occurs 170 days after January 1. Except for leap year, this corresponds to June 20. The shortest day occurs 353 days after January 1, or December 20.