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.
The water level in a pool is given by a quadratic function \(h(t) = at^2 + bt + c\), where \(t\) is measured in hours.
At the moment when the pool is completely drained, say at time \(t_0\), we have \(h(t_0) = 0\) and \(h'(t_0) = 0\).
It is also known that after the first hour, the water level has dropped to exactly half of its original value: \(h(1) = \tfrac{1}{2} h(0)\).
How many hours does it take for the pool to drain completely?