Why does the death of a living being affect the decay of carbon-14?

Carbon-14 is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. Carbon-14 decays into nitrogen-14 through beta decay:

By emitting a beta particle (an electron, e^-) and an electron antineutrino (ν_e), one of the neutrons in the carbon-14 nucleus changes to a proton and the carbon-14 nucleus becomes the stable (non-radioactive) isotope nitrogen-14.

The equation governing the decay of a radioactive isotope is

$$ N=N_0 e^{-\frac{t}{\tau}}$$

where N_o is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. On the other hand, the mean-life τ is the average or expected time a given atom will survive before undergoing radioactive decay.

Since the amount of carbon-14 inside a piece of wood or a fragment of bone decrease as the carbon-14 undergoes radioactive decay, measuring the amount of carbon-14 in a sample provides information that can be used to calculate when the animal or plant died. The mean-life of carbon-14 is 8267 years, so the equation above can be rewritten as:

Nevertheless, radioactive decay is a process that takes place inside the nucleus, so nor a change of temperature neither chemical reactions affect radioactive decay. Carbon-14 atoms inside a living being are decaying after and before the living being dies. So why is this method used efficiently to measure when the living being died? How do we know N_o, the amount of carbon-14 the living being had at the moment it died, if carbon-14 was also decaying when the plant or the animal was alive?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

The Moon is getting further away from Earth. Where does this extra energy come from?

All bounded orbits where the gravity of a central body dominates are elliptical in nature. In the case of the Moon orbiting the Earth, the eccentricity of the ellipse is so small (0.055) that it is almost a circle:

Therefore, the gravitational force F_g that the Earth exerts on the Moon is perpendicular to Moon's velocity v, so it is a centripetal force F_c, making the trajectory of the Moon bend:
$$ F_{g}=F_{c} \\
\frac{GMm}{r^2}=\frac{mv^2}{r} $$ where G is Newton's constant, M is Earth's mass, m is Moon's mass and r is the radius of the orbit.

This implies that the kinetic energy of the Moon is
$$
K=\frac{1}{2}mv^2=\frac{GMm}{2r}
$$ which is smaller than the absolute value of the potential energy
$$
U=-\frac{GMm}{r}
$$ So the mechanical energy of the Moon is
$$
E=-\frac{GMm}{2r}
$$

We know that at the time of its formation, the Moon sat much closer to the Earth, a mere 22,500 km away, compared with the 402,336 km between the Earth and the Moon today. So the Moon is getting further away from Earth, now at the rate of 3.78 cm per year. Nevertheless, according to the last equation, a larger r means that the Moon has more energy every year. Is its energy non conserved? Who is giving energy to the Moon?