Ant and RectangleDate: 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/ |
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