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Problem #PRU-111923

Problems Calculus Derivative Derivative and extrema Derivative and multiple roots

15–16 3.0

When water is drained from a pool, the water level $h$ in it varies depending on the time $t$ according to the function $h (t) = a t^{2} + b t + c$ , 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?

Problem #PRU-65325

Problems Calculus Derivative Derivative and extrema Probability and statistics Probability theory Discrete distribution Increasing and decreasing. Researching functions

15–16 4.0

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.