This is a fairly challenging problem from James multi-variable calc book 5 ed. Here it is: A cow is tied to a silo with radius r by a rope just long enough to reach the other side. What is the total grazing area?
I've spent quite a good amount of time working on it and come up with some good ideas, but I've hit a bit of a stump, and I'm not getting past it.
I'm sorry I don't have a scanner, this would be of some use for the diagrams but hopefully this will be enough. I divided the area in two parts (and subdivided those) cutting through the silo. One of those parts of the top half, A1 say, is just a quarter circle of radius pi(r). The other half on the other hand looks a bit like a quarter of a cardioid. This will be called A2. A3 is the area of the top half of the silo. 2(A1+A2-A3)=A
So now I need a model for A2. I started with the semi-circle here and 0<t<pi x=r(1-cost) and y=rsint. I also figured the length of the rope "against" the silo would be (pi)r-tr. Last of all the tangents of the points on the circle should tell me which direction the rope will be and as I know what the distance left is I figured I could extract the coordinates. (y'=cost/sint). And this is where I'm stuck. I tried using the distance formula here (which was a pain) to find an equation for the coordinates and the x could not be isolated. After coming this far, I'd like to finish, but I'm afraid I may have not picked the best way of representing this. Any insight appreciated.