An aquaculture farm operates a shrimp population that evolves according to natural reproduction and harvesting. The initial mass of this shrimp population is estimated at 100 tonnes. Given the reproduction and harvesting conditions, the mass of the shrimp population, expressed in tonnes, as a function of time, expressed in weeks, is modelled by the function $f _ { p }$, defined on the interval $[ 0 ; + \infty [$ by :
$$f _ { p } ( t ) = \frac { 100 p } { 1 - ( 1 - p ) \mathrm { e } ^ { - p t } }$$
where $p$ is a parameter strictly between 0 and 1 and which depends on the various living and exploitation conditions of the shrimp.
- Model consistency a. Calculate $f _ { p } ( 0 )$. b. Recall that $0 < p < 1$.
Prove that for all real number $t \geqslant 0,1 - ( 1 - p ) \mathrm { e } ^ { - p t } \geqslant p$. c. Deduce that for all real number $t \geqslant 0,0 < f _ { p } ( t ) \leqslant 100$.
2. Study of evolution when $p = 0.9$
In this question, we take $p = 0.9$ and study the function $f _ { 0.9 }$ defined on $[ 0 ; + \infty [$ by :
$$f _ { 0.9 } ( t ) = \frac { 90 } { 1 - 0.1 \mathrm { e } ^ { - 0.9 t } }$$
a. Determine the variations of the function $f _ { 0.9 }$. b. Prove that for all real number $t \geqslant 0 , f _ { 0.9 } ( t ) \geqslant 90$. c. Interpret the results of questions 2. a. and 2. b. in context.
3. Return to the general case
Recall that $0 < p < 1$. Express as a function of $p$ the limit of $f _ { p }$ as $t$ tends to $+ \infty$.
4. In this question, we take $p = \frac { 1 } { 2 }$. a. Show that the function $H$ defined on the interval $[ 0 ; + \infty [$ by :
$$H ( t ) = 100 \ln \left( 2 - \mathrm { e } ^ { - \frac { t } { 2 } } \right) + 50 t$$
is an antiderivative of the function $f _ { 1/2 }$ on this interval. b. Deduce the average mass of shrimp during the first 5 weeks of exploitation, that is the average value of the function $f _ { 1/2 }$ on the interval $[ 0 ; 5 ]$. Give an approximate value rounded to the nearest tonne.