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.
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?
To find the constants a and b so that S(x) is continuous, we need to analyze the piecewise function:
For 0 ≤ x < x0, S(x) = a + b * arcsin(tan(x) / tan(x0))
For x0 ≤ x ≤ 90, S(x) = 24
To make S(x) continuous, we need to make sure that the two parts of the function match at x = x0.
At x = x0, we have:
S(x0) = a + b * arcsin(tan(x0) / tan(x0)) = a + b * arcsin(1) = a + b * (π/2)
Since S(x0) = 24, we can set this equal to 24:
a + b * (π/2) = 24
Now let's calculate S(x) for Tucson, Arizona (x = 32° 13'):
Using the formula for 0 ≤ x < x0:
S(x) = a + b * arcsin(tan(x) / tan(x0))
S(32° 13') = a + b * arcsin(tan(32° 13') / tan(x0))
Similarly, let's calculate S(x) for Walla Walla, Washington (x = 46° 4'):
S(x) = a + b * arcsin(tan(x) / tan(x0))
S(46° 4') = a + b * arcsin(tan(46° 4') / tan(x0))
Lastly, to graph S(x), we'll plot S(x) for each value of x from 0 to 90, taking into account the two parts of the piecewise function.
To determine if S(x) is differentiable, we need to check if it is continuous and if its derivative exists at all points in the interval. Since we have established S(x) as a continuous function, we can now analyze if its derivative exists for all values of x in the interval 0 ≤ x ≤ 90.