Question

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.

Answer 1

b.
S= a+b(tanx/tanxo)
for S(0)
S(0)=0= a+b*0 so a=0
Now for s(x0)=24
S(x0)=24=a+b(tan xo/tanxo)
or b=24
The real question is S(x) differentiable. Examine the point at x0. Is the curve continous at that point?