Scenario: Farmer Jones had a goat on a tether. He tied the end of the tether not attached to the goat to a stake in a field. Over what area could the goat graze?

There are two structures in the field:

-- a shed that is 20 ft long and 20 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 center to center 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.

Here is an expression for this scenario:

Case 1: If the stake is in the midpoint of the barn then what is the area the goat can graze?Here is a GSP file which demonstrates the area where the goat can graze if the stake is in the midpoint:

Case 2: Now what is the area the goat can graze if the stake is at the outer edge of the silo?

Here is a GSP file for this case:

Calculations:

1) The total area where the goat can graze:

If we consider the circle with radius 76.7 ft then the area of this circle is approximately 18472.2746 ft square.

The area of shed is 400 ft square.

The area of silo is approximately 314.2857 ft square.

Area of the rectangle between silo and shed is 14112.8 ft square.

So the total area will be less than 18472 + 14112 - 314.2857-400 = 31869. 7143 ft sqaure. I said the total area will be less than 31869.7143 ft square because when we attached tether outher edge of silo and shed then the goat will not graze all of the area. You can see this from below GSP 3.

2) Maximum area: The goat graze maximum area when we attach tether between silo and midpoint and it will be approximately:

18472.2746 - 314 - ( the area goat cannot graze) = 18158.2746 - ( the area goat cannot graze)

So it will be less than 18158.2746 ft square.

Conclusion:

In conclusion we can find the area where the goat can graze by using case1 and case 2. It is obvious that the maximum area the goat can graze when the stake in between silo and midpoint because the area of shed is larger than area of silo so the stake must be close to silo in order to get maximum area. Now what is the total area for the goat graze?

Here is a GSP file which demonstrates an area for the goat can graze:

Note: I calculate the areas approximately so to get exact results we must find the missing areas where the goat cannot graze.