The goat can graze 4847.071825 square feet.
Consider the following diagram, for purposes of discussion.
The area the goat can graze is the sum of the red, blue, and yellow regions.
It is important to solve for the angle ADC, which is atan(sqrt(2)/sqrt(34)) =~ 0.237941125 radians.
Forgive me if I don't get into the details, but here are the areas of each region, by color:
Red = 3769.911184
Blue = 921.005359
Yellow = 156.155281
Total = 4847.071825
This question is asked and discussed in my forum at Wizard of Vegas.