```Question 535885
Something is missing. I don't know where the fence is with respect to the barn. If the corner the goat is tied to is 50 feet or more from the fence in all directions, the goat can reach the end of its rope and the fence does not matter. If the goat can hit the fence before the rope is taut, then the grazing area will be smaller. As a friend of the goat, this is my design for barn, fence, an tether point. The large rectangle represents the fence, the small one the barn.
{{{drawing( 750, 600, -70, 80, -60, 60,
rectangle(-65, -55, 75, 55 ),
rectangle(-20, 0, 20, 20 ),
red( circle( 20, 0, 50 ) ),
green( circle( 20, 20, 30 ) ),
blue( circle( -20, 0, 10 ) ),
locate( 20, 0, A ), locate( -30, 0, B ), locate( 20, 50, C ),
locate( 70, 0, x ), locate( -20, 10, D ), locate( -10, 20, E ),
line( 20, 20, 20, 50 ),
line( -20, 0, -30, 0 )
)}}} The goat, tied at point A, can walk from X past C to E in one direction.
Turning the other way, it could walk from X through B to D.
In this best case (for the goat), the goat will have access to three fourth of the 50-foot circle (red) around the tether point. The remaining section of that circle is partially occupied by the barn. moreover, the goat will be able to walk along the side and past the end of the barn with rope to spare, and not hit the fence. The goat will have 10 feet of rope left past the end of the long side, and 30 feet left past the end of the short side. At each of the corners of the barn closest to the tether point, the goat will be able to turn around, along part of the adjacent side, and sweep another quarter circle of grazing area. In that best case, the grazing area will be 3/4 of a circle of radius 50ft (red), plus one 1/4 of a circle of radius 10 ft (blue), plus 1/4 of a circle of radius 30 ft (green).
{{{area=(3/4) (pi) 50^2+ (pi) 10^2/4+ (pi) 30^2/4=(pi/4)(3*2500+100+900)=(pi/4)*8500=6675.88}}}
```