Lake Covering Problem

A circular lake of radius 10 meters must be covered by rectangular boards of length 30 meters.
Rectangular boards of all widths are available, but the cost of a covering is the total width of the boards used. Clearly the lake can be covered by using parallel boards of total width 20 meters.
Is there a more efficient covering possible? Here are examples of two competing coverings.


Solution to Particles Riddle

In a previous post I discussed the Particles riddle, a solution of which is as follows. Cover the unit square by four closed subsquares of side length 1/2, denoted A, B,C,D as in the figure. Suppose that the initially awake particle is located at x_A \in A, and there are sleeping particles in B,C,D (the case when some of these are unoccupied is easy to handle.) The particle at x_A wakes up a particle at some point x_B \in B by time \sqrt{5/4}. During the next \sqrt{2} time units, one of these two particles can travel to wake a particle in some x_D in D, and at the same time, the other  can travel to wake a particle at some x_C in C. In another 1/2 time unit, one of the particles from x_D can travel to B and one of the particles at x_C can travel to A. Thus by time \sqrt{5/4}+\sqrt{2}+1/2<3.1, there will be an awake particle in each of the originally occupied subsquares A, B,C,D. The argument can now be iterated in each of these subsquares. Repeated subdivision will yield a geometric series 3.1(1+1/2+1/4+\ldots)=6.2 as an upper bound for the time needed to reach all n particles. The constant 6.2  can certainly be improved; we will not try to optimize this constant. See also a similar solution posted by Sang-il Oum here.