In a certain city, number of hours of sunlight on summer solstice is 15.848 and
ID: 3398883 • Letter: I
Question
In a certain city, number of hours of sunlight on summer solstice is 15.848 and the number of hours of sunlight on winter solstice is 8.454.Assume summer solstice occurs on the 172nd day of the year, abd thatthere are 365 days untim the next summer solstice.A)find sinusoidal function B) use function found on part A to predict number of hours of sunlight on april 1, the 91st day of year In a certain city, number of hours of sunlight on summer solstice is 15.848 and the number of hours of sunlight on winter solstice is 8.454.Assume summer solstice occurs on the 172nd day of the year, abd thatthere are 365 days untim the next summer solstice.
A)find sinusoidal function B) use function found on part A to predict number of hours of sunlight on april 1, the 91st day of year
A)find sinusoidal function B) use function found on part A to predict number of hours of sunlight on april 1, the 91st day of year
Explanation / Answer
solution:
according to calender
June 21 is the 172nd day of the year, and December 21 (the winter solstice) is the 355th day of the year
if we assume 1 jan isfirst day of the year
now the time period should be 365 days
if i assume function
y=sin(B(t-C))+D
Since the range of t is from 1 to 365, and the range of the input should be 0 to 2*pi
B(365 - C) - B(1 - C) = 2*pi - 0
B * (365 - C - 1 + C) = 2*pi
B * 364 = 2*pi
B = 2*pi / 364
now
summersolistic means maximum daylight means 15.848 maxima at x=pi/2
(2*pi / 364) * (172 - C) = pi / 2
172 - C = 364 * pi / (2 * 2*pi)
172 - C = 364 * pi / (4*pi)
172 - C = 364 / 4
172 - C = 91
C = 172 - 91
C = 81
Now B and C are calculated now calculating A and D by putting maximum daylight at x=pi/2
and minimum daylight at x=-pi/2
15.848= A sin [2*pi / 364 * (172 - 81)] + D
15.848 = A sin [2*pi / 364 * 91] + D
15.848 = A sin [pi / 2] + D
15.848 = A * 1 + D
15.848 = A + D ..................(i)
and at minima
8.454 = A sin [2*pi / 364 * (355 - 81)] + D
8.454 = A sin [1.505 * pi] + D
now
sin [1.505 * pi] is approximately -1
8.454 = D-A ............(ii)
adding (i) and (ii)
D=12.151
A=3.697
now our sine function is
y=3.697sin[2*pi / 364 * (t - 81)]+12.151
b) now at t=91
y=3.697sin[2*pi / 364 * (91 - 81)]+12.151
y=3.697sin [0.1726]+12.151
y=3.697*0.1718+12.151
y=12.7861hours