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POW Solution, January 3-7
Posted:
Jan 10, 1994 11:52 AM
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******************************
Problem of the Week for January 3-7
A camper is off picking berries. He is strolling along,
bucket full of berries in hand, when he sees that his
tent is on fire. Luckily, he is near a river, so he
can run to the river, fill his bucket with water, and
run to the tent to put the fire out. The question:
Where should he go along the bank of the river should
he fill the bucket to make his trip as short as
possible and have the greates chance of saving his tent?
X
Camper with bucket
\
\ Y
\ Possible route _______/ Burning tent
\ ______________/
\ __________/
\ ____/
______\/___________________________________________________
River
__________________________________________________________
Of course, the route back to the tent should be a straight line.
******************************
"Correct" Solutions were received from:
Nick Szmyd Grade 10, Shaler Area High School
Bipin Mujumdar Grade 10, Shaler Area Senior High School
Alternative solutions, which aren't exactly what we were "mathematically"
looking for, but certainly aren't incorrect, were submitted by
Larureanna Raymond, 10th grade, Edgerton High School, Edgerton, WI
Willy Deutschbauer and Chuck Raida, Grade 12, Shaler Area High School
Both "kinds" of solutions are included below.
******************************
From: nick szmyd <szmyd@one.sasd.k12.pa.us>
*In starting this problem, I first reflected X over the river
and called that point Z. That made the distance from X to the river
equal to the distance from Z to the river. I then connected point Z
to point Y because the shortest distance between two points is a
straight line. In doing that, I found the point at which ZY
intersects the river and called that point M. I then found out,
that with the river being the perpendictular bisector of XZ, that XM
equals ZM by the perpendictular bisector theorem. Since the shortest
path between two points is a straight line, and segment MY+ZM =ZY, I
just added segment MY to segment XM. The path of XM+MY equals the
path of ZM+MY. So, in answering the question, point M is where the
man should run to the river.
* This answer is based upon the river being straight and the
area obstacle free.
Nick Szmyd
Grade 10
Shaler Area High School
******************************
From: bipin mujumdar <mujumdar@one.sasd.k12.pa.us>
At the start of the problem I assumed the following:
1) The river was perfectly straight and did not curve around the tent
2) The problem wished for the shortest distance not necessarily the
shortest time
3) The river was as narrow as a simple line
4) The man ran in perfectly straight lines
To find the point to fill the bucket I reflected the point where
the man started over the river which I considered a line. In doing
this, the line becomes the perpendicular bisector of the segment
formed by the original and reflected point.Since the shortest
distance between two points is a straight line, I drew in the line
from the reflected point to the burning tent. I concluded that the
intersection of that line and the river was the point at which the
man should fill his bucket. My reasoning was that segments drawn from
the original man with bucket to the intersection were equal to the
reflected point to the intersection because of the perpendicular
bisector theorem. I finally reasoned that the path taken would equal
the straight line from the reflected point to the tent were equal
because the sides were equal and if you add Z to two equal number,
the results will also be equal. In this case Z is equal to the
distance between the intersection and the tent. Since both of the
final line was equal and one of them was a straight line, the
intersecting point was where the man should fill his bucket.
Bipin Mujumdar, Shaler Area Senior High School, tenth grade
******************************
From: adam deutschbauer <deutschb@one.sasd.k12.pa.us>
Date: Fri, 7 Jan 94 14:03:43 -0600
To: pow@forum.swarthmore.edu
Subject: problem of the week
The camper should head to a point on the river directly south of the
burning tent. This way he can run for the longer distance with no
water in his pail. Thus, the camper can travel the shortest distance
with a full pail of water. Therefore he will have less chance to
spill the water.
Willy Deutschbauer and Chuck Raida, Grade 12, Shaler Area High School
******************************
From: akuemmel@students.wisc.edu (Andrew Kuemmel)
Solution by Larureanna Raymond, 10th grade, Edgerton High School, Edgerton, WI
X
Camper with bucket
\
\ Y
\ Possible route _______/ Burning tent
\ ______________/ |
\ __________/ |
\ ____/ |
______\/________________________________________|___________
River Z
Find the line perpendicular to the river through the tent. This is the
shortest distance from the river to the tent. You would want to follow
this route because you can carry a lot of water with the least chance of
spilling it.
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