by

Rene T. Estomo

Tethered Goat

The investigation is to solve the problem below.

Problem:

Farmer Jones had a goat on a tether. He tied the end of the tether not attached to the goat to a stake in the field. Over what area could the goat graze? Of course you need to know something about the length of the tether and about the field.

There are two structures in the field:

---- a shed that is 20 ft long and 29 feet wide (square)

---- a silo that is 20 ft in diameter

The center of the shed and the center of the silo are on a line and the distance apart is 92 feet. The distance from the center, if you wanted this data, is 112 feet.

The tether for the goat is 76.7 feet long. The stake to which the tether is tied is somewhere along the line of centers between the shed and the silo.

Explore the area over which the goat can graze as the stake is moved along this line segment from the midpoint of the side of the shed to the edge of the silo.

There are some areas grazed by the goat which are easy to solve. These areas are shown below.

The next area which is exciting to solve is the one above the shed. Below is my exploration.

The last area is challenging to compute. This is the one around the silo.

Now, we already have completed all the possible grazing areas of the goat. So, the total MINIMUM grazing area of the goat is

= A1 + A2 + A3 + A4 + A5

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