Interesting word problem posed

[twinhairdryer]

So I had a student today ask me if I could come up with a simple 2 dimensional area problem. I was like -sure what is it. The question was:

If a cow were strapped to a tether outside a 10' by 10' square barn, at the corner of a small outlet door, and had a 100' leash/tether, what would the area be that he could cover to graze?

So the obvious is a circular sweep CCW with a 100 square foot keep out area in one direction, but then other? any ideas how to compute this one? thanks newbie

 

[Dick]

It not a 'simple' problem. It's a complicated problem. Not theory-complicated, just technical-complicated. You'll have to figure out the cow's path with the tether stretched to it's max CCW and CW. At some point they'll probably intersect. Then add up all the area of all the arc sectors making up the path up to that point of intersection.

 

[HallsofIvy]

It's best to start by drawing a picture. Obviously the cow can graze in three-fourths of a circle of radius 100, the three quadrants not containing the barn. Now stretch the tether across the front of the barn. You can now think of the cow as tethered at the other corner and the tether is now 100-10= 90 feet long. The cow can graze in one-fourth of a circle of radius 90. Now move to the third corner. The tether is now 90- 10= 80 feet long but you've already included a portion of this quadrant in your first three-quarters of a circle. The "new" grass is the portion of a circle of radius 80 "behind" the barn: between the lines formed by the sides of the barns extended. That is equal to the area bounded by the lines x= 0, x= 10, y= 0 and the curve x²+ y²= 80².

Added in edit: oops. Just as Dick said, if the cow goes around the barn the other way, she will pick up a little bit more grass symmetric to that last part. Because it is symmetric do this: find the area bounded by the lines x= 0, x= 5, y= 0 and x²+ y²= 80[/sup]2[/sup] and double it to get both parts.

Wrong again! It is not symmetric. Going around the barn the other way, the cow only has to pass the side of the barn, not front and side. The point where the two circles meet in back will be the intersection of x²+ y²= 80² and (x-10)²+ y²= 90². And you will need to find the area bounded by x= 0, x= a, where a is the x coord on that intersection, y= 0 and x²+ y²= 80², find the area bounded by x= a, x= 10, y= 0 and (x- 10)²+ y²= 90², and add.

 

[Dick]

Actually, there is a usable symmetry. The path is symmetric across the diagonal connecting the corner of the barn the cow is tethered to and the opposite corner. That helps.