When was sunrise? |
 |
Solution 1 by Alex Bordyukov, Grosse Pointe South High School
First let x represent the time of sunrise, p the speed of
Pat, and d the speed of Dana. It took Pat 5 hours to get to town B
after he crossed paths with Dana, and it took 11.25 hours for Dana to get to
town A after she crossed paths with Pat; and it took each of them hours 
to get to the crossing point. That gives us two equations: 
and 
. Now solve
the second equation for d, and plug it into the first equation. You’ll
get the equation 
:,
which can be rewritten as 
.
When we solve for x, we get two answers, 19.5 and 4.5. The answer can’t
be 19.5 because that would make a
negative value; so the answer is that sunrise was at 4:30 am.
Solution 2 by Steve Colvin, West Bloomfield High School
When Pat and Dana cross paths, they have each traveled an equal amount of time,
which can be called x. It took Dana 11.25 hours to traverse the distance
Pat did in x hours, and it took Pat 5 hours to traverse the distance
Dana did in x hours. Because each person walks at a constant speed,
the ratios of their times are the same. So, we can write an equation to find
x as 
.
Cross-multiplying shows 
,
and 
. Therefore,
both Pat and Dana set out 7.5 hours before noon. This means that sunrise occurred
at 4:30 am.
Also solved by: Joshua Schlegel (Bedford H.S.); Andrew Jeanguenat (De La Salle Collegiate H.S.); Peter Denton (East Grand Rapids H.S.); Hyunje Grace Cho, Alex Kim (Grosse Pointe South H.S.); David Guo, Michelle Khadra (Troy Athens H.S.); Joe Meleca, Jay Yang (Forest Hills Central H.S.); Joe Davila, Mischa Dylewski, Nikkel Westdorp (Forest Hills Eastern H.S.); Grant Boyer, Chase Schuler (Forest Hills Northern H.S.); Matthew Brownell, Armila Francis, Kristen Kitti, Aaron Lorincz, Patrick Lutz, Melissa Rich, Sara Ross (Plymouth Christian Academy); Trevor Lambert, Jake Maloney (Detroit Catholic Central H.S.); Walter Wang (Plymouth H.S.); Taylor Thurston, Zac Warner (Linden H.S.); Josie Keenan, Kiran Sebastian, Alex Timm (Saginaw Heritage H.S.); and Sara White (Gettysburg H.S., Gettysburg, PA).