In a laboratory, a chemical reaction is studied in a closed reactor under certain conditions. The numerical processing of experimental data made it possible to model the evolution of the temperature of this chemical reaction as a function of time. The objective of this exercise is to study this modeling. Temperature is expressed in degrees Celsius and time is expressed in minutes. Throughout the exercise, we are on the time interval $[ 0 ; 10 ]$. Parts A and B can be treated independently.
Part A
In an orthogonal coordinate system of the plane, we give below the representative curve of the temperature function as a function of time on the interval $[ 0 ; 10 ]$.
- Determine, by graphical reading, after how much time the temperature returns to its initial value at time $t = 0$. [Figure]
We call $f$ the temperature function represented by the curve above. We specify that the function $f$ is defined and differentiable on the interval $[ 0 ; 10 ]$. It is admitted that the function $f$ can be written in the form $f ( t ) = ( a t + b ) \mathrm { e } ^ { - 0,5 t }$ where $a$ and $b$ are two real constants. 2. It is admitted that the exact value of $f ( 0 )$ is 40. Deduce the value of $b$. 3. It is admitted that $f$ satisfies the differential equation $( \mathrm { E } ) : y ^ { \prime } + 0,5 y = 60 \mathrm { e } ^ { - 0,5 t }$.
Determine the value of $a$.
Part B: Study of the function $f$
It is admitted that the function $f$ is defined for every real $t$ in the interval $[ 0 ; 10 ]$ by
$$f ( t ) = ( 60 t + 40 ) \mathrm { e } ^ { - 0,5 t }$$
- Show that for every real $t$ in the interval $[ 0 ; 10 ]$, we have: $f ^ { \prime } ( t ) = ( 40 - 30 t ) \mathrm { e } ^ { - 0,5 t }$.
- (a) Study the direction of variation of the function $f$ on the interval $[ 0 ; 10 ]$.
Draw up the table of variations of the function $f$ by including the images of the values present in the table. (b) Show that the equation $f ( t ) = 40$ admits a unique solution $\alpha$ strictly positive on the interval $] 0 ; 10 ]$. (c) Give an approximate value of $\alpha$ to the nearest tenth and give an interpretation of it in the context of the exercise. 3. We define the average temperature, expressed in degrees Celsius, of this chemical reaction between two times $t _ { 1 }$ and $t _ { 2 }$, expressed in minutes, by
$$\frac { 1 } { t _ { 2 } - t _ { 1 } } \int _ { t _ { 1 } } ^ { t _ { 2 } } f ( t ) d t$$
(a) Using integration by parts, show that
$$\int _ { 0 } ^ { 4 } f ( t ) d t = 320 - \frac { 800 } { \mathrm { e } ^ { 2 } }$$
(b) Deduce an approximate value, to the nearest degree Celsius, of the average temperature of this chemical reaction during the first 4 minutes.