Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.

A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.

In this part, for any non-negative real $t$, we denote $\theta(t)$ the temperature of the coffee at instant $t$, with $\theta(t)$ expressed in degrees Celsius and $t$ in minutes. Thus $\theta(0) = 80$.

In this model, more precise than that of part A, we assume that $\theta$ is a function differentiable on the interval $[0; +\infty[$ and that, for any real $t$ in this interval, Newton's law is modeled by the equality:
$$\theta'(t) = -0{,}2(\theta(t) - M).$$

\begin{enumerate}
  \item In this question, we choose $M = 0$. We then seek a function $\theta$ differentiable on the interval $[0; +\infty[$ satisfying $\theta(0) = 80$ and, for any real $t$ in this interval: $\theta'(t) = -0{,}2\theta(t)$.\\
  a. If $\theta$ is such a function, we set for any $t$ in the interval $[0; +\infty[$, $f(t) = \frac{\theta(t)}{\mathrm{e}^{-0{,}2t}}$. Show that the function $f$ is differentiable on $[0; +\infty[$ and that, for any real $t$ in this interval, $f'(t) = 0$.\\
  b. Keeping the hypothesis from a., calculate $f(0)$. Deduce, for any $t$ in the interval $[0; +\infty[$, an expression for $f(t)$, then for $\theta(t)$.\\
  c. Verify that the function $\theta$ found in b. is a solution to the problem.
  \item In this question, we choose $M = 10$. We admit that there exists a unique function $g$ differentiable on $[0; +\infty[$, modeling the temperature of the coffee at any non-negative instant $t$, and that, for any $t$ in the interval $[0; +\infty[$:
$$g(t) = 10 + 70\mathrm{e}^{-0{,}2t},$$
where $t$ is expressed in minutes and $g(t)$ in degrees Celsius.

A person likes to drink their coffee at $40^{\circ}\mathrm{C}$.\\
Show that there exists a unique real $t_0$ in $[0; +\infty[$ such that $g(t_0) = 40$.\\
Give the value of $t_0$ rounded to the nearest second.
\end{enumerate}