We are told that the longest possible ride is a straight line L1 of length 200 feet. Such a ride must start and stop on the outer fence, and it must just graze the inner fence. Symmetry shows that the point of contact P is exactly halfway along the 200 foot line.
Any line that just grazes a circle is a tangent line to that circle; this means that a line from the center of the circle will be perpendicular to the tangent line. Now, denote the inner radius by r and the outer radius by R. Consider the diameter line L2 that passes through the center of the circle and the point of tangency P.
The point P divides L1 into two segments, each of length 100. It divides L2 into two segments of length R-r and R+r. It is a geometric fact that if two lines intersect inside a circle, the product of the lengths of the two segments of one line is equal to the corresponding product of the lengths of the two segments of the other line.
This implies:
100 * 100 = ( R - r ) * ( R + r )
10,000 = R*R - r*r
Now, note that the area of the track can be expressed as
A = pi*R*R - pi*r*r
= pi * ( R*R - r*r )
= pi * 10,000
= 31416 square feet, approximately.
This is how Little Bob figured out how much asphalt was needed, even though he still doesn't know the actual sizes of the inner and outer circles!
Oddly enough, if you know that Little Bob was able to work out the area of the track, then you can skip all the math and go directly to the solution. There are a number of different track sizes which have the same property of a maximum 200 foot ride in them. If Little Bob can figure out the area correctly despite this fact, then all such tracks must have the same area. Let's pick the simplest such track: the inner radius r is 0, and the outer radius R is 100. This track has an area of pi*R*R, or about 31416 square feet. So, if the problem is soluble, then this must be the answer! Sometimes the lazy man finds the quickest way.
Back to The Parking Puzzle.