Grazing for Mooey

A farmer tied his cow, Mooey, to a post with a rope that allowed Mooey to be a maximum of 5 m away from the post. How many square meters of grazing ground does Mooey have?

If the farmer moves Mooey to another field and ties her to a post with the same rope at the corner of a closed shed 3 m by 3 m in the field (see diagram--again, Mooey's maximum distance from the shed is 5 m), how many square meters of grazing ground does Mooey have?

Mooey can hurt herself if she is capable of going all the way around the barn. What is the longest safe length of rope that can be tied to Mooey so she has the greatest possible grazing area? Explain.

If you watch a dog tied to a post with a pivot bolt, it moves around the post in a circlular fashion.  So the shape I will be working with is a circle.  Using the area formula for a circle and 3.14 for pi, I find that the area is 78.50meters squared. (I realize this counts the area occupied by the post.  I was not given knowledge of the size of the post). The picture is scaled so that 1m equals 1 cm.  To see Mooey enjoying her pasture, click here.

Unfortunately for me, this part became much more complicated than I expected.  I did not know the shape of the shed so I assumed it was a quadrilateral. Using that knowledge, I drew a shed with 4 sides of 3 meters each.  This shed being a square seems to be the most obvious solution.  I have not seen many rhomboidal sheds.

My first thought in finding MooeyÕs grazing area was to subtract 9 (the area of the shed) from the answer to the preceding problem.  However, this was too simplistic.  The corner of the shed will block Mooey from reaching the grass in her former grazing area. Instead, once she reaches the corner of the shed, her ability to move changes.  Instead of subtracting the just the area of the shed, I must subtract an entire quarter of the circle. (I know it is a quarter because of the definition of square which must have right angles.)  So I take one quarter of the 78.50 which is 19.63 and subtract it from 78.50.  This results in 58.88 meters square.  That is the definite grazing area for Mooey.

Once Mooey gets to an edge of the shed her rope hangs her up.  Lets consider when she gets to one edge.

Now if she tried to go clockwise, the corner of the shed has become another pivot point.  The remainder of her rope is the radius of the circle she can walk.  Since her original rope is 5 meters and the length of the side of the shed is 3meters, she has two more meters of play.

MooeyÕs path using the corner of the shed as a pivot point is a semi circle.  However, after traveling one half of this semicircle, she runs into the wall of the shed.  To computer this area, I use 2 as the radius, multiply by3.14 and multiply by .25 to yield 3.14 meters squared. Since there are two corners Mooey must navigate around, I multiply 3.14 times 2.

I add this result to my previous result and get an area of 65.61 meters squared.

Mooey can hurt herself if she is capable of going all the way around the barn. What is the longest safe length of rope that can be tied to Mooey so she has the greatest possible grazing area? Explain.

Since going around the shed is the problem, the maximum length of the rope is less than 12meters which is the perimeter of the shed. I personally would subtract the length of MooeyÕs head (which at this time is unknown and varies with the age of Mooey) from the perimeter of the barn.