Question

"A plane contains points A and B with AB=1. Let S be the union of all disks of radius 1 in the plane that cover AB. What is the area of S?"

As taken from the 2004 AMC. Here's a math challenge for those of you who like math. ;)

Answer 1

oh dang... there are just way too many circles to graph till u get to the outermost (union of all of them)

Answer 2

and then u gotta subtract some complements of the intersection, uhmm no thanks!

Answer 3

lol

Answer 4

Too many circles to graph? Nonsense. You have two points connected by a line, the circles just pivot on each one.

Answer 5

And then you have the circles that must lie on both A and B.

Answer 6

I'm not redrawing it here, but you have two sectors with area 4pi/3, two sectors with area pi/6, and the intersection has area sqrt(3)/2. Answer should be 8pi/3+pi/3-sqrt(3)/2=3pi-sqrt(3)/2

Answer 7

But he's asking for a union of everything

Answer 8

i see six minimum

Answer 9

Right you are, Sir! Here, have several solutions.

Answer 10

S is the union of all the disks, but it asks for the area of S, not the area of all the disks.