Ponder This Challenge - February 1999 - A leashed sheep eats grass

Ponder This Challenge:

There is a circle of grass with the radius R. We want to let a sheep eat the grass from that circle by attaching the sheep's leash on the edge of the circle. What must be the length of the leash for the sheep to eat exactly half of the grass? This problem was sent in by Mirek Wilmer. FYI: Although this problem may appear simple at first, the solution is actually fairly complex.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  Mainstream solution: Let L denote the length of the leash. Let O be the center of the grass circle, and Q the location where the leash is fastened. Let P and P' be the two points on the circumference of the grass circle at distance L from Q.

  Let B denote the measure of angle PQO in radians, and (C = π - 2B) the measure of POQ. Because PQO is isosceles, we have L = 2 R cos B. The pie-shaped region emanating from O and reaching from P to P' has area (1/2) R2 (2C) = R2 C.

  The pie-shaped region emanating from Q and reaching from P to P' has area L2 B. Together these regions cover the sheep's eating area, but they both cover the quadrangle OPQP', so we must subtract its area, 2 ( (1/2) R L sin B) = R L sin B.

  We obtain ( R2 C ) + ( L2 B ) - R L sin B = (1/2) π R2, from which ( R2 ( π- 2B))+( 4 R2 B cos2 B )-( 2 R2 sin B cos B )=(1/2)π R2, or π - 2B + 4 B cos2 B - 2 sin B cos B = π /2. We solve this numerically for B, and obtain B = 0.952848, C = 1.235897, L=1.158728R.

  Alternate solution: Impossible. Even if we ignore the possibility of extraterrestrial causes of crop circles, the circle of grass can only have arisen because of an irrigation device which pivots in the center of the circle. The sheep (being a stupid animal) will get tangled up in the irrigation mechanism, and will never finish eating his half of the grass.

Solvers