At a certain latitude in the northern hemisphere, the number d of hours of daylight in each day of the year is taken to be d=A+Bsinkt°, where A, B, k are positive constants and t is the time in days after the spring equinox.
Assuming that the number of hours of daylight follows an annual cycle of 365 days, find the value of k, giving your answer correct to 3 decimal places.
Ok I'm not really sure where to start here, any hints would be appreciated.
Am I supposed to use -1<sinkt°<1 ? how to i fit in the 365 day with t?
The following should give you a start as to how to tackle your specific problem.
How many hours of sunlight does Florida receive during the summer solstice depending on latitude?"
Lets make a picture first representing the earth as seen by you looking at it from space on the summer solstice with the Sun to the right..
1--Draw yourself a 2 - 3 inch diameter circle with center O representing the earth.
2--Draw a line through O from the lower left to the upper right at 23.5º to the vertical extending the line outside the circle.
3--Label the upper end of this line N for north and the lower end S for south.
4--Draw a diameter of the circle through O at right angles to the line NS labeling the left end A and the right end B. This line represents the equator.
5--To the right of your circle write a big S indicating the direction from which the Sun's rays are coming from.
6--Draw a vertical line through O to the circle's circumference which represents the terminator line of the Sun's rays on the earth, the dividing line between daylight and darkness, labeling the upper end E and the lower end F.
7--Draw a line from point O at 28.5º to OB and above line AB to point D on the circumference. Point D represents a point in Florida at 28.5º North Latidude, approximately Cape Canaveral. (/_DOB = £ = 28.5º)
8--Draw another line from D, parallel to, and above, AB to the circle's opposite circumference labeling the left end C, representing the circle of 28.5º north latitude.
9--Label the point of intersection of NS with CD point G.
10--Label the point of intersection of EF with CD point H.
11--Let OB = R, the earth's radius, ~3963 miles.
12--Let OG = a, GD = b, and HG = c.
13--a = Rsin(£) = 3963(.4771) = ~1891 miles
14--b = Rcos(£) = 3963(.8788) = 3483 miles.
15--c = a(tan(23.5º) = 1891(.4348) = ~822 miles.
Now imagine a plane through NS, perpendicular to the paper on which this picture is drawn. Imagine you are looking down on the North pole with your eyes in that plane.
16--Point H represents the point on the earth's surface at 28.5º north latitude that is transitioning from darkness to daylight. What we want to find out is what angle point H makes with this NS plane perpendicular to the picture.
16--Looking down on the north pole, a line OH represents the line from the earth's center to the terminator line at 28.5º north latitude.
17--This line makes an angle µ with the NS plane, the magnitude of which is arcsin(c/b) which in our case is µ = arcsin(822/3843) = 12.35º.
18--Therefore, daylight begins at a point 12.35º behind the NS plane and daylight ends at a point 180 + 2(12.35º) away or 204.7º.
19--This number of degrees between sunrise and sunset translates into the number of hours of daylight which is then DL(hr) = 204.7(24)/360 = 13.64 hours.
20--Putting the above together, the daylight hours are
DL(hr) = [180 + 2arcsin{.4348Rsin(£)/RcosL}]/15 where L = the degrees of north latitude.
Considering the north latitude of New York City of 40.714º, it sees 14.93 hours of daylight on the date of the
summer solstice.
Considering the arctic circle at 66.5º north latitude, anyone on the circle would see 24 hours of daylight.
With the figures developed here, see what you have to do to arrive at your answer.
These numbers do not take into account the fact that the earth rotates slightly more than 360º in a 24 hour solar day. Taking this into account would add only minutes to the length of the daylight period.
