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### Ant and Rectangle

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Date: 01/22/2001 at 23:38:51
From: Li
Subject: The ant and the rectangle

Here is my question:

An ant walks inside a 18cm by 150 rectangle. The ant's path follows
straight lines that always make angles of 45 degrees with the sides of
the rectangle. The ant starts from a point x on one of the shorter
sides. The first time the ant reaches the opposite side, it arrives at
the midpoint. What is the distance, in centimeters, from x to the
nearest corner of the rectangle?

Does that mean the ant is walking along the diagonals of the
rectangle? That way the answer would be too obvious.

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Date: 01/23/2001 at 16:20:36
From: Doctor Greenie
Subject: Re: The ant and the rectangle

The ant doesn't walk along the diagonals of the rectangle. Walking in
straight lines that make 45 degree angles with the sides of the
rectangle means that in crossing from one side of the rectangle to the
other, the ant will move a distance down the rectangle equal to the
width of the rectangle, which is 18 cm.

If the ant ends up at the midpoint of the opposite side from where he
started, then he ends up 9 cm from each end of that side. This means
he last touched one of the long sides of the rectangle 9 cm from the
end of the rectangle.

Now let x be the number of cm from a corner where the ant started;
then the first time he touches the long side of the rectangle, he is
either x cm or (18-x) cm from the end of the rectangle where he
started. If n is the number of times he crosses from one side of the
rectangle to the other, then we must have

x + 18n + 9 = 150

where n is an integer.

Then you have to remember that the question asks for the distance from
where the ant started to the nearest corner, so if solving the
equation above gives the answer x = 12, then the answer to the problem
is 6 cm (if he started 12 cm from one corner, then he was 6 cm from
the other corner....)

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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