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?

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Request for Question Clarification by answerguru-ga on 25 Nov 2005 08:10 PST

This is a duplicate - you may want to remove it so you aren't charged twice.

answerguru-ga

Umm, what a nice challenge. 
It is locked on the other posting so this one is okay.

Let x = distance by trail from A to B towns.
And P = rate or constant speed of Pat.
And D = constant speed of Dana.
And t = time at sunrise, based from 12:00 midnight.

distance = speed *time

Pat and Dana met at 12:00 noon. That is 12hrs from midnight.
That means the distances travelled by the two in (12-t) hours are
equal to the total distance from A to B.
x = P(12-t) +D(12-t)
x = (P+D)(12-t)  --------(1)

Pat reached B at 5:00 PM. That was 17hrs from midnight.
That means the distance travelled by Dana from sunrise to noon is the
same in distance as what Pat travelled from noon to 5PM.
P(17-12) = D(12-t)
P = D(12-t)/5  ---------(2)

Dana reached A at 11:15 PM, or 23.25 hrs from midnight.
That means Dana covered the whole distance from A to B in (23.25 -t) hours.
x = D(23.25 -t)  --------(3)

Substitute the P from (2) into (1),
x = [D(12-t)/5 +D](12-t) 
x = (D/5)(17-t)(12-t)
Plug in there the x from (3),
D(23.25-t)= (D/5)(17-t)(12-t)
Divide both sides by (D/5),
5(23.25-t) = (17-t)(12-t)
116.25 -5t = 204 -29t +t^2
0 = t^2 -29t +204 +5t -116.25
t^2 -24t +87.75 = 0
Using the Quadratic Formula,
t = {24 +,-sqrt[(24^2) -4(1)(87.75)]} /(2*1)
t = (24 +,-15)/2
t = 19.5 or 4.5

19.5 is not okay, so t = 4.5 or 4:30 AM.
That was the sunrise. 

Very early sunrise. Is that right?
Let us see.

Plug t=4.5 into (1),
x = (P+D)(12-4.5) = 7.5(P+D)  -----(i)

Into (2),
P = D(12-4.5)/5 = 1.5D  ------(ii)
That into (i),
x = 7.5(1.5D +D) = (7.5)(2.5D) = 18.75D  ------(iii)

Plug t=4.5 into (3),
x = D(23.25 -4.5) = 18.75D  ------same as (iii)

Therefore, sunrise at 4:30 AM is okay.

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 ) :)