Little-known fact The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude 40° north, the latest sunrise occurs on January 4 at 7:25 A.M. (14 days after the solstice), and the earliest sunset occurs on December 7 at 4:37 P.M. (14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4:30 A.M. (14 days after the solstice) and the latest sunset occurs on June 7 at 7:32 P.M. (14 days before the solstice). Using sine functions, devise a function s ( t ) that gives the time of sunrise t days after January 1 and a function S ( t ) that gives the time of sunset t days after January 1. Assume that s and S are measured in minutes and s = 0 and S = 0 correspond to 4:00 A.M. Graph the functions. Then graph the length of the day function D ( t ) = S ( t ) − s ( t ) and show that the longest and shortest days occur on the solstices.
Little-known fact The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude 40° north, the latest sunrise occurs on January 4 at 7:25 A.M. (14 days after the solstice), and the earliest sunset occurs on December 7 at 4:37 P.M. (14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4:30 A.M. (14 days after the solstice) and the latest sunset occurs on June 7 at 7:32 P.M. (14 days before the solstice). Using sine functions, devise a function s ( t ) that gives the time of sunrise t days after January 1 and a function S ( t ) that gives the time of sunset t days after January 1. Assume that s and S are measured in minutes and s = 0 and S = 0 correspond to 4:00 A.M. Graph the functions. Then graph the length of the day function D ( t ) = S ( t ) − s ( t ) and show that the longest and shortest days occur on the solstices.
Solution Summary: The author calculates the length of the day function and shows that the longest and shortest days occur on the solstices.
Little-known fact The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude 40° north, the latest sunrise occurs on January 4 at 7:25 A.M. (14 days after the solstice), and the earliest sunset occurs on December 7 at 4:37 P.M. (14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4:30 A.M. (14 days after the solstice) and the latest sunset occurs on June 7 at 7:32 P.M. (14 days before the solstice). Using sine functions, devise a function s(t) that gives the time of sunrise t days after January 1 and a function S(t) that gives the time of sunset t days after January 1. Assume that s and S are measured in minutes and s = 0 and S = 0 correspond to 4:00 A.M. Graph the functions. Then graph the length of the day function D(t) = S(t) − s(t) and show that the longest and shortest days occur on the solstices.
The radian of a number system Equals the number of its distinct counting digits why?
What time is indicated on the clocks
when the Sun passes due south on
the 21st November in a City (43°N
75°W)? A Coruña uses Central
European Time, with the standard
time zone meridian at 15°E. Daylight
saving time is not in effect in winter.
Calculate the declination and right ascension of the
sun when it is at longitude 90° (summer solstice) and
longitude 45°. By symmetry, find the declination at
longitudes 270° and 315°.
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