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? 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 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, if you wanted to use 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.

If the tether that the goat is tied to can move on the line between the silo and the shed, then the green figure has enclosed the area that the goat can reach. Below is a calculation of this enclosed area.

To find the area of the possible grazing area, I will break the area into three separate parts.

1. The area between the silo and the shed

The area in the middle is easiest to find so we will start here. This area is simply rectangular field that measures 92 feet by 153.4 feet. Thus the area here is 14112.8 square feet.

2. The area around the shed

We need to find the grazing area around the barn. This is a little more complicated. First, let’s make an estimate about an upper bound on the area. We know that the area here is less than the semicircle formed with a radius of 76.7 feet (this is the radius from the middle of the barn to edge shown in blue) reduced by the area of the barn. Remember the area of the barn is 20x20=400. Thus we know it is less than:

Thus we know the grazing here must be less than 8840.82 square feet.

Now we can make a more accurate approximation for the area enclosed by the green figure. This will be done by finding the area of the individual sectors. First we can look at the sectors called sector 1. We know these sectors are each ¼ of a circle with a radius of 66.7 feet. Thus we can find the area of these by solving:

Remember there are two of these sectors, so their total are is double 3494.15.

Second, we need to find the area of the triangle. We know the sides of the triangle are 46.7 feet and the base is 20 feet. Since the triangle is isosceles, the altitude is also the perpendicular bisector of the triangle. This means we can find the height of the altitude by solving the Pythagorean Theorem, where the altitude is b:

Now we can find the area of the triangle follows the formula:

Finally we need to find the areas of the remaining sectors, called sector 2. Again, we know the length of the arc is 46.7 (the same as the side of the triangle). We need the angle of the arc and we know this is 90°-θ, where θ is the angle of the isosceles triangle side. We can find this angle using trigonometry, as we know cosθ=adjacent/hypotenuse. Hence, cosθ=10/46.7=0.2141 so θ=77.64°. Now we know the angle of the arc is 90°-77.64°=12.36°.

When we convert 12.36° to radians, we see that the angle is 0.0343pi.

Then we can use the area formula below:

Remember there are two of these sectors, so their total are is double 235.32 square feet. So the total area here is 470.64 square feet.

Now we can conclude that the area that the goat can graze near the barn is sum of all of these pieces: Since this area is less than 8840.82 square feet, we know it should be accurate.

3. The area around the silo

We need to find the area the goat can graze around the silo. We will again being by making an upper bound for this area. Again this is the area of the semicircle reduced by the area of the silo (this is shown in blue). Here the area of the silo is 100 square feet. Thus the area here should be less than 8926.66 square feet.

To see a GSP file of how to construct the area arond the silo please click here.

To find the area here, we will make an approximation as calculating the actual area will be very difficult. We will do this by finding approximate lengths of sectors and adding them up.

First, we will look at sector 1. The green segment was drawn to mark the edge of this sector. This was drawn so that it matches the GSP file of how the tether will wrap around the silo. Using GSP, the length of this segment is 69.5 feet. To estimate this area, we want one eight of the circle with a radius of 69.5 feet.

Thus we can solve:

So, the area here is 1896.83 square feet.

Second, we will look at sector 2. The pink segment now shows the edge of this sector. Again, GSP was used to estimate this length. The length was found to be 61.82 feet. We can then estimate this area by finding one eight of the circle with a radius of 61.82 feet. Thus we can solve:

Hence, the area here is 1500.78 square feet.

Third, we can move to sector 3. The length of this sector is the purple line. This length is the same as the pink line’s length since the tether will still get stuck at the same point. Thus the length is 61.82. GSP was then used to calculate the angle length of the sector. This was found to be 8.92 degrees. This needed to be converted to radians though:

So the angle is 0.1556 radians.

Now we can find the area of this sector:

Finally, we need to find the area of the triangle that can be reached. Since the triangle is isosceles, the altitude is also the perpendicular bisector of the triangle. This means we can find the height of the altitude by solving the Pythagorean Theorem, where the altitude is b:

Given the altitude is 61.01 feet, we can now find the area of the triangle:

The whole triangle cannot be reached by the goat though. Half the area of the silo must be removed from it since there is no grass where the silo is:

Thus the goat can only graze 295.94 square feet here.

To find the total area the goat can graze by the silo, we must add up sector 1, sector 2, sector 3, and the area of the triangle less half the area of the silo. Since this is only a calculation for part of the field, all the sectors must be doubled. Hence our result is:

Since our result is less than the maximum allowed, it should be a good approximation.

**Solution:**

So now we know the total grazing area for the goat is 29713.38 square feet.

Complete a Write-up on your Web Page for one additional investigation, chosen from Assignment 0 through Assignment 12. Clearly it should be an investigation other than one you have written up before.

Selected problem: Assignment 12 number 8. Please click here for this part of the assignment.