Two towns, A and B, are located along the Appalachain Trail.  At
sunrise, Pat begins walking south from A to B along the trail, while
simultaneously Dana begins walking north from B to A.  Each person
walks at a constant speed, and they cross paths at noon.  Pat arrives
in B at 5pm while Dana reaches A at 11:15pm. When was sunrise?

Let x be the distance from the noon meeting point to A.
Let y be the distance from the noon meeting point to B.

The rate at which Pat travels is:
y/5
The rate at which Dana travels is:
x/(11 1/4) = 4x/45

Let t be the time from sunrise to noon in hours.

From sunrise to noon, Pat traveled distance x. This distance is his
rate of speed times the time from sunrise:
x=(y/5)t
From sunrise to noon, Dana traveled distance y. This distance is his
rate of speed time the time from sunrise:
y=4xt/45

Solving these two equations for t:
t=5(x/y)
t=(45/4)(y/x)

The right hand sides both equal t, so

5(x/y)=(45/4)(y/x)

Multiplying both sides by (x/y) and by 1/5 gives:
(x/y)^2 = 9/4

x/y=3/2
Substituting in  the first equation for t gives:

t=5(3/2)= 7.5 hours

Sunrise is 4:30 am.

OK, I'm conflicted.

First, I saw that the question had been locked, perhaps twice. That
got me curious about the problem, even though I suspected the question
might be a homework problem.

Second, I realized that there are three unknowns, at least the way I
look at the problem - so, if there is an answer, two of the three
unknowns must be in a constant ratio.  That made it an interesting
problem.

Third, this question could have been locked again any time, so I put
my solution in as a comment _before_ researching via google.

After posting, I googled the phrase "begins walking south from A to B
along the trail" and found that this is a homework problem. (What it
has to do with calculus, I don't know.) But it's just fun (at least if
you are under no pressure to solve the problem) to think about what
information is available and what to do when there is no readily
available formula.

So, in an attempt to redeem myself, here is how I thought about the
problem: I drew a line with A and B at both ends and marked a point
labeled "noon meeting point." (I generally like to draw something or
write down the information given in summary form; I think better about
a problem if I get all the information collected in one of these two
ways.) Then I labeled one distance x and the other y. The only times
given are the times to go from the noon meeting point to opposite
ends, so one can get the rates of travel of both Pat and Dana (but in
terms of x and y). Then I let t be the time from sunrise to noon. At
this point, I swithed to looking at the distances traveled before
noon; I had both x and y as distances again but no specific time - no
5 or 11 1/4 hours - but we know that the times to the noon meeting
point are equal. After that, I looked for equations with t and found I
could solve for the x/y ratio.

The key for me was to draw a diagram, then look at the part of the
problem with more information (travel after noon), then go back and
look at travel before noon. I hope that this might help you for more
than this problem.

They met at noon 12.
After that Pat walked 5 hours Dana walked 11:15 hours ; so the way is 16:15 hours.
Pat arrived B at 17 (5pm).
17-16:15=0.45
sunrise was at 00:45 am ( the city should be around Alaska ) :)

I don't think you can just add the hours since the 5 hours was at one
rate of speed and the 11.25 hours was at another rate of speed. The
two distances covered in the 5 hours and in the 11.5 hours add up to
the total distance, but I think you can add the hours only if two
distances are covered at the same rate of speed.