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August 21, 2014

August 21, 2014

Posted by **Sara** on Friday, November 12, 2010 at 7:11pm.

Hours of Daylight as a Function of Latitude.

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

a.) What is S(0)?

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

My answers:

a.) @ 0 degrees latitude, you're at the equator, so S(0) = 12 hours daylight

b.) Since S(0)=12,

12 = a + b arcsin * (tan0/tan x0)

12 = a + b arcsin 0

12 = a + b0

12 = a,

and for b, because the function must be continuous,

24 = 12 + b arcsin* (tan 66/tan 66)

12 = b arcsin* 1

7.639 (rounded) = b.

But now when I try to find the answer to part C,

latitude of Tucson= 32 degrees

S(x)= 12 + 7.639 arcsin* (tan 32/tan 66)

...I get a nonreal answer for the arcsin of (tan 32/tan 66).

Can anyone please tell me where I'm going wrong? Thanks!

- Calculus -
**MathMate**, Friday, November 12, 2010 at 7:45pm"I get a nonreal answer for the arcsin of (tan 32/tan 66)"

"arcsin of (tan 32/tan 66)" exists and equals 16.15338661° as long as the angles 32 and 66 are in degrees. I guess that your calculator has been set to radians or grad. Please check.

Post if you have other queries.

I am in the process of going through your other answers, so unable to comment for now.

- Calculus -
**Sara**, Saturday, November 13, 2010 at 2:21pmOh, I am sorry! I had my calculator on radian mode instead of degree mode. Silly me. Thanks so much for the help!

I think I have managed to do the remainder of the problem, except for part E. How would I be able to tell if that function were differentiable or not?

- Calculus -
**Andreas**, Friday, October 7, 2011 at 10:44amHow would you draw the graph of this problem?

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