In a large and flat grass field, there is a 30x30 feet square barn facing due North. Straight from the NorthEast corner of the barn is a fence running due East 40 feet long. There is a horse tied on the SouthEast corner of the barn with a rope that will allow him to eat grass in any direction up to 100 feet. The horse can't eat under the barn, and the rope can't pass through the barn or through the fence. The horse can walk around the barn or the fence, but is limited by the length of the rope.

How many square feet of grass can the horse reach?

(I found this problem pretty interesting, but I am not the author and have no idea who is the author).

The easiest way to look at this is as a series of circle-sectors and a triangle.

(1), there are two quarter-circles with radius 100 (southeast, southwest of the hitching point ): 2(1/4)ð(100)²

(2), a quarter-circle of radius 70 (northwest from the southwest corner of the barn (1/4)ð(70)²

(3) a quarter-circle of radius 40 (northeast from the northwest corner of the barn) This terminates against the fence 10 feet from the northeast corner of the barn (and the western end of the fence) and 20 feet from the eastern end of the fence: (1/4)ð(40)² Going the other way,

(4) you add the area of the 30-40-50 triangle formed by the two ends of the fence and the southeast corner of the barn: 1/2(30)(40)

(5) Add the 100 foot radius sector northeast of the southeast corner up to the point where the rope touches the fence: (100)²arctan(3/4)

(6) Add the sector of radius 50 from the endpoint in the last step to the point 50 feet north of the eastern end of the fence: (50)²arcsec(3/4)

(7) Add the quarter-circle of radius 50 northwest of the eastern end of the fence, ending along the north face of the barn 10 feet west of the northeast corner and 20 feet east of the northwest corner: (1/4)ð(50)²

(8) Subtract the overlap between sectors (3) and (7): ???

Posted by TomM on 2005-04-03 04:17:07