Ant and Rectangle

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. 

Thank you for your time.

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/