SOLUTION: A farmer has fenced off his trapezoidal housing area, which is shown in the diagram. There is a post at each of the point A and B, to which the farmer sometimes attaches a 30m rope

Question 1205305: A farmer has fenced off his trapezoidal housing area, which is shown in the diagram. There is a post at each of the point A and B, to which the farmer sometimes attaches a 30m rope that is tied to his donkey. This provides the donkey either one of two grazing areas outside the housing area. Find the difference in the areas available for grazing in square meters (m^2)

Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!

Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Here's a hint:

If the donkey is tethered to point A, then this is his/her grazing area

I used GeoGebra to make the diagram.
There are 3 color-coded regions.
Each region corresponds to a different pizza slice of sorts (i.e. a circle sector).
I apologize if the diagram seems a bit cluttered.

Region 1, in blue, is the largest because it has the largest radius (30 meters) and largest central angle (240 degrees).

Let's say the donkey tries to go as far as he or she can to the southwest direction. At some point the rope will run along side of the house's outer wall along segment AB. The furthest point southwest the donkey can go is point J.

If the donkey wanted to go directly south of point A, s/he will have to venture into region 2 (red).
The central angle of region 2's pizza slice is 120 degrees. The radius is 30-16 = 14 meters.
The red region is partially equivalent to having the donkey tethered to point B with a 14 meter leash.

At this point I should mention that the formula you'll use to find the area of each pizza slice is:

The portion is the area of the full circle of radius r, and the part out front is us taking a fractional part of this entire circle.
The angle must be in degrees.

Region 3 has its central angle of 90 degrees and it has a radius of 30-20 = 10 meters (since AD = 20 meters eats up that much rope to leave 10 meters for the donkey to swing around the right side of the house).
The green region is partially equivalent to having the donkey tethered to point D with a 10 meter leash.

I'll leave the calculations for the student to do.

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If you tether the donkey's leash to point B, then this is the grazing area.

I'll let the student determine the central angles for each pizza slice.
Note how region 3 is very small, but still part of the donkey's grazing area.
The radius of region 3 is 30-28 = 2 meters.

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