|Have you seen this movie? I liked it.|
Norville has four different colored hula hoops around his waist, each with a diameter of 3 feet. He is standing on a flat surface. He drops a red hula hoop so that he is standing exactly in its center. He then walks 5 feet, stops, and drops a blue hula hoop (again, so that he is standing in its center). Norville then walks 10 feet in a different direction, and drops a green hula hoop in the same way. Finally, he turns and walks 8 feet in another direction before dropping his last hula hoop, a purple one, in the same manner he dropped the others. What is the greatest possible area of overlap the purple hoop can have with another hoop, and with which hoop can it have that overlap?You can't answer anonymously to win, but don't like leave your address in the comments or anything—we will take care of the shipping arrangements once I declare a winner. If you live outside the US, you can still win, but you are going to have to help me cover shipping costs. Sucks, I know.
Also, I want answers in terms of π. Don't give me any decimal approximations.
UPDATE: Congratulations to "Sjfour4," whose book is en route! Solution below the cut.
I love this question because like many SAT questions, it's not nearly as complicated as it seems at first glance. The first thing you should be asking yourself is whether Norville is able to complete a triangle with his 3 short trips. In other words, is it possible to have a triangle with sides of length 5, 10, and 8?
To evaluate the possibility, check against the triangle inequality theorem:
Is 10 + 5 > 8? YES.
Is 10 + 8 > 5? YES.
So that triangle is possible. Here's an artist's rendition of Norville's trip (using cm instead of feet because that's how my software measures things):
The diameter of each hula hoop is 3, so the radius of each is 1.5. That means the area surrounded by each hoop is π(1.5)2 = 2.25π.
Since the triangle can be completed, the purple hoop can be dropped right on top of the red hoop, and the overlap can be the entire area covered by the hoop: 2.25π.