THE TETHERED GOAT.
(
Patchwork Puzzles)
Here is a little problem that everybody should know how to solve. The
goat is placed in a half-acre meadow, that is in shape an equilateral
triangle. It is tethered to a post at one corner of the field. What
should be the length of the tether (to the nearest inch) in order that
the goat shall be able to eat just half the grass in the field? It is
assumed that the goat can feed to the end of the tether.
Answer:
This problem is quite simple if properly attacked. 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
11/2 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."
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