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 \(40^{\circ}\) 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.
