When water is drained from a pool, the water level \(h\) in it varies depending on the time \(t\) according to the function \(h (t) = at^2 + bt + c\), and at the time \(t_0\) of when the draining is ending, the equalities \(h (t_0) = h' (t_0) = 0\) are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?
Hercules meets the three-headed snake Hydra of Lerna. Every minute, Hercules chops off one head of the snake. Let \(x\) be the survivability of the snake (\(x > 0\)). The probability \(p_s\) of the fact that in the place of the severed head will grow s new heads \((s = 0, 1, 2)\) is equal to \(\frac{x^s}{1 + x + x^2}\).
During the first 10 minutes of the battle, Hercules recorded how many heads grew in place of each chopped off one. The following vector was obtained: \(K = (1, 2, 2, 1, 0, 2, 1, 0, 1, 2)\). Find the value of the survivability of the snake, under which the probability of the vector \(K\) is greatest.