Answer :
Let us suppose the triangle ABC to represent our
half-acre field, and the shaded portion to be the quarter-acre over
which the goat will graze when tethered to the corner C.

Now, as six equal equilateral triangles placed together will form a
regular hexagon, as shown, it is evident that the shaded pasture is
just one-sixth of the complete area of a circle.

Therefore all we require is the radius (CD) of a circle containing six
quarter-acres or 1½ acres, which is equal to 9,408,960
square inches.

As we only want our answer "to the nearest inch," it is sufficiently
exact for our purpose if we assume that as 1 is to 3.1416, so is the
diameter of a circle to its circumference.

If, therefore, we divide the last number I gave by 3.1416, and extract
the square root, we find that 1,731 inches, or 48 yards 3 inches, is
the required length of the tether "to the nearest inch."