Posted by **Gerald** on Friday, October 10, 2008 at 5:13am.

Hours of Daylight as a Function of Latitude

Let S (x) be the number of sunlight hours on a cloudless June 21, as a function of latitude, x,

measured in degrees.

(a) What is S (0)?

(b) Let x0 be the latitude of the Arctic Circle (x0

≈ 66◦ 30′ ). In the northern hemisphere, S (x)

is given, for some constants a and b, by the formula:

S (x) = bracket (piecewise function)

a + b arcsin*(tan x/tan x0)

for 0 ≤ x < x0

24 for x0 ≤ x ≤ 90.

Find a and b so that S (x) is continuous.

(c) Calculate S (x) for Tucson, Arizona (x = 32◦ 13′ ) and Walla Walla, Washington (46◦ 4′ ).

(d) Graph S (x), for 0 ≤ x ≤ 90.

(e) Does S (x) appear to be differentiable?

I have the following answers:

a) What is S(0)? You're at the equator, so the S(0)=12.

b) S(x) is at the artic circle which, as noted above, gets 24 hours of sunlight on June 21st. S(x) = 24.

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