posted by Geni .
Can someone please explain this to me?? I'm really stuck on it. First of all, i don't know if I should use cosine or sine for the function because it only says to make a sinusoidal function that models the data, and a sinusoidal function could be either sine or cosine. PLEASE HELP!!!
Trigonometric Stuff: It is possible to use a sinusoidal function to model the amount of perceived daylight in a certain location over the course of a year. For Portland, Oregon, there is a minimum of 9 hours of “daylight” on the 1st day of winter and a maximum of 17 hours of “daylight” on the 1st day of summer. Let D represent the number of hours of “daylight” in Portland, Oregon, T days after the 1st day of spring (assume that T = 0 corresponds to March 20th). You may assume that 1 year has 365 days.
Find a formula for such a function, being sure to explain the practical meanings of any important pieces of the formula (amplitude,midline, and period). Use your formula to determine on what days of the year (month and day, not just T’s value) Portland has about 11hours of “daylight” and about 15 hours of “daylight”. Please round to the nearest day, if not exact.
Please can someone help!?!!?
well, you could say
D = .5(9+17)+ A sin wt + B cos wt
that is the midline , 13
if you start at the minimum with t = 0
and D = 9
then to get a minimum at t = 0
say D = 13 + A sin 0 + B cos 0
D = 13 + 0 + B
but we know D = 9
so B = -4
so we have
D = 13 - 4 cos w t
now when t = 365
w t = 2 pi, a whole circle
w * 365 = 2 pi
w = 2 pi/365
d = 13 - 4 cos (2 pi t/365)
it starts at the median (average) day length and increases to the summer solstice, then decreases to the winter solstice, then increases back to the average at the vernal equinox
the midline is the average ... (9 + 17) / 2
the amplitude is max minus average
the period is 365 (days)
whoops I took t = 0 at Jan 1
should be at March 20 or 1/4 year after Jan 1
alter that to get min at 1/4 year
D = 13 - 4 cos (2pi t/365 - x)
when t = (1/4)(365) or 91.25 days
2 pi * .25 - x = 0
x = 2 pi * .25
so in the end
D = 13 - 4 cos (2 pi t/365 - pi/2)
D = 13 + 4 sin (2 pi t/365)
Thank you so much!!! It makes more sense now!!! Thanks again!!