The Problem: Elizabeth tethered her goat Sadie to one end of a 20-foot length of rope and attached the other end of the rope to a hook fixed to the side of her shed. When she came back Sadie had eaten all the grass within its reach. Where did Elizabeth fix the rope? What was the area available to the goat? Could the goat have eaten more grass if the rope had been attached to a different point on the shed? Experiment with different lengths of rope. Where should the hook be positioned each time to allow the goat to eat the maximum amount of grass?

Start by drawing a good diagram. Use graph paper, and label the diagram with any information you have.

We know the rope is 20 ft long. We can use this knowledge, plus the measurements of the building to figure out where the rope is tied. Because the goat can only get to the top right corner, we know he must be tied on the bottom side of the shed. Because the side of the shed is 10 ft long, and the bottom is 15, we know the goat is tied 5 ft from the bottom left corner. Once we know where the rope is tied, we can use that information to determine the length of the various areas… 10 ft 10 ft

Area in red:

Area in blue- ¼ of a circle: Area in green- ¼ of a circle:

Area in orange- ¼ of a circle: Total area available to the goat: 200π + 25π + 56.25π + 6.25π = 287.5π = (287.5)(3.14)= 902.75 ft2 200π 25π 56.25π 6.25π