DP 5-3
Discovery Project ■ Hours of Daylight


Objective To model the hours of daylight at a given latitude with a sinusoidal function.

The number of hours of daylight (sunrise to sunset) varies from day to day throughout the course of a year. In the Northern Hemisphere the longest day (Summer Solstice) is around June 21, and the shortest day (Winter Solstice) is around December 21. The average length of daylight is 12 h, and the variation from this average depends on the latitude. For example, Fairbanks, Alaska (latitude 65° N), experiences more than 20 h of daylight on the longest day and less than 4 h on the shortest day! However, at places close to the equator there is far less difference between the shortest and longest days of the year.The graph in Figure 1 shows the number of hours of daylight from March° 21 through December 21 at several different latitudes. It’s apparent from the graph that the variation in hours of daylight is simple harmonic and so a model for the number of hours of daylight is a function of the form $L(t)$=$a$ sin$(w(t-c))$+$b$.

Five curves are graphed on a coordinate plane on the grid lines. The horizontal axis is displayed with months from March to December. The vertical axis labeled Hours is displayed from 0 to 20 in increments of 2 hours. The purple curve which is located at approximately 60 degrees North latitude begins at 12 hours on March 21, reaches its approximate peak at 18.5 hours on June 21, then falls to 12 hours on September 21, and ends with an approximate low point at 5.5 on December 21. The yellow curve which is located at approximately 50 degrees N latitude begins at 12 hours on March 21, reaches its approximate peak at 16.2 hours on June 21, then falls to 12 hours on September 21, and ends with an approximate low point at 7.8 on December 21. The blue curve which is located at approximately 40 degrees North latitude begins at 12 hours on March 21, reaches its approximate peak at 14.8 hours on June 21, then falls to 12 hours on September 21, and ends with an approximate low point at 9.2 hours on December 21. The red curve, which is located at 30 degrees North latitude, begins at 12 hours on March 21, reaches its approximate peak at 14 hours on June 21, then falls to 12 hours on September 21, and ends with an approximate low point at 10 hours on December 21. The black curve, which is located at 20 degrees North latitude, begins at 12 hours on March 21, reaches its approximate peak at 13 hours on June 21, then falls to 12 hours on September 21, and ends with an approximate low point at 11 hours on December 21.
Figure 1 Graph of the length of daylight from March 21 through December 21 at various latitudes
EXAMPLE 1 | Modeling the Number of Hours of Daylight

In Philadelphia (40° N latitude) the longest day of the year has 14 h 50 min of daylight, and the shortest day has 9 h 10 min of daylight.

  1. Find a function $L$ that models the number of hours of daylight as a function of $t$, the number of days from January 1. (So then $t$ = 1 corresponds to January 1.)
  2. An astronomer needs at least 11 hours of night for a long exposure astronomical photograph. On what days of the year are such long exposures possible?

SOLUTION

  1. We need to find a function of the form
    $L(t)$ = $a$ sin$(ω(t-c))$+$b$
    whose graph is the 40° N latitude curve in Figure 1 (the blue curve), where $L$ is the number of $hours$ of daylight and $t$ is the number of days since January 1. Since $L$ is measured in hours, we see from the information given that the longest day has $14\frac{5}{6}$ hours of daylight and the shortest day has $9\frac{1}{6}$ hours of daylight. The amplitude $a$ of the sine curve is the average of these two numbers.
    $a=\frac{1}{2}(14\frac{5}{6}-9\frac{1}{6})\approx 2.83h$
    Since $t$ is measured in days, and there are 365 days in a year, the period is 365, and so
    $ω =\frac{2π}{365}\approx 0.0172$
    As noted in the introduction, the average number of hours of daylight is 12 h, and so the graph is shifted upward by 12 units and $b$ = 12. Since the curve attains the average value (12) on March 21, the 80th day of the year (in 2023), the curve is shifted 80 units to the right, and so $c$ = 80. Thus a function that models the number of hours of daylight is
    $L(t)$ = $2.83$sin$(0.0172(t-80))$+$12$
    where $t$ is the number of days from January 1
  2. A day has 24 hours, so 11 hours of night correspond to 13 hours of daylight. To find all the days that have at most 13 hours of daylight, we need to solve the inequality ${y \le 13}$. To solve this inequality graphically, we graph the functions
    $y$ = $2.83$sin$(0.0172(t-80))$+$12$ and y=13
    on the same graph. From the graph in Figure 2 we see that there are fewer than 13 hours of daylight between day 1 (January 1) and day 101 (April 11) and between day 241 (August 29) and day 365 (December 31). So the astonomer can make a 11-hour long exposure photograph between August 29 of one year to April 11 of the following year.
    A curve and a horizontal line are graphed on a graphing calculator window screen. The horizontal axis is displayed from 0 to 365 in increments of 121 units. The vertical axis is displayed from 0 to 15 in increments of 1 unit. The curve starts approximately from (0, 10), goes up to the right, creates a high, and begins to go down to the right and exit the right of the viewing window of the graphing calculator approximately at point (365, 10). The horizontal line starts from point (0, 14), goes to the right passes through the curve at point t = 101 and t = 241, and exits the top right viewing window.
    Figure 2

Problems

  1. Modeling the number of hours of daylight in your town
    Let’s find a sine curve that models the number of hours of daylight at your town, starting on January 1. First, search the Internet (for instance go to www.timeanddate.com) to find the number of hours of daylight for the longest and shortest days of the year in your town.
    1. Longest day has _____________ hours of daylight.
    2. Shortest day has _______________ hours of daylight.
    3. The amplitude of the sine curve is $a$ = _______________.
    4. The period is 365, so $ω= \frac{\color{red}{\fbox{  }}}{\color{red}{\fbox{  }}} \approx$ ________.
    5. Since the average length of a day is 12 h, the vertical shift is $b$ = ________.
    6. The curve attains the average value on ________, and so the horizontal shift is $c$ = ________.
    7. A function that models the number of hours of daylight on day $t$ (where $t$ is the number of days from January 1) is
      $y$ = ${\color{red}{\fbox{  }}}$sin$({\color{red}{\fbox{  }}}(t-{\color{red}{\fbox{  }}}))$+${\color{red}{\fbox{  }}}$
    8. What day of the year is today?________. How many days is this date from January 1 of this year? _________. To use the model you found to estimate the number of hours of daylight for today set $t$ _____. So from the model we find that the number of hours of daylight for today is _____. Does your answer agree with your Internet search for the number of hours of daylight in your town today?