A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
where $C$ denotes the concentration of the medication in the blood (in micromoles per liter), $t$ the time elapsed since the start of the infusion (in hours), $d$ the infusion rate (in micromoles per hour), $a$ a strictly positive real parameter called clearance (in liters per hour).
Part C: determination of appropriate treatment
The purpose of this part is to determine, for a given patient, the value of the infusion rate that allows the treatment to be effective, that is, the plateau to equal 15. The infusion rate $d$ is provisionally set to 105.

1. We seek to determine the clearance $a$ of a patient. The infusion rate is provisionally set to 105. a. Express as a function of $a$ the concentration of the medication 6 hours after the start of the infusion. b. After 6 hours, analyses allow us to know the concentration of the medication in the blood; it is equal to 5.9 micromoles per liter. Determine an approximate value, to the nearest tenth of a liter per hour, of the clearance of this patient.
2. Determine the value of the infusion rate $d$ guaranteeing the effectiveness of the treatment.

Organochlorine pesticides were widely used in agriculture; however, due to their high toxicity and persistence in the environment, they were banned. Consider the application of 500 g of an organochlorine pesticide to a crop and that, under certain conditions, the half-life of the pesticide in the soil is 5 years.
The mass of pesticide over 35 years will be closest to
(A) $3.9 \mathrm{~g}$.
(B) $31.2 \mathrm{~g}$.
(C) $62.5 \mathrm{~g}$.
(D) $125.0 \mathrm{~g}$.
(E) $250.0 \mathrm{~g}$.

A researcher analyzed the data on the number of new cases of a disease in a city over a period of 5 consecutive years and organized them in the table below.

Year	New cases
1	200
2	400
3	800
4	1600
5	3200

Based on this data, the researcher modeled the number of new cases as a function of the year $x$ by the expression $f(x) = 100 \cdot 2^x$.
If this trend continues, in which year will the number of new cases first exceed 100,000?
(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

8. The shelf life $y$ (in hours) of a certain food and storage temperature $x$ (in ${ } ^ { \circ } \mathrm { C }$ ) satisfy the functional relationship $y = e ^ { k x + b }$ (where $e = 2.718 \ldots$ is the base of natural logarithm, and $k , b$ are constants). If the shelf life of this food at $0 { } ^ { \circ } \mathrm { C }$ is 192 hours and at $22 { } ^ { \circ } \mathrm { C }$ is 48 hours, then the shelf life at $33 { } ^ { \circ } \mathrm { C }$ is
(A) 16 hours
(B) 20 hours
(C) 24 hours
(D) 28 hours

13. The shelf life $y$ (in hours) of a certain food and the storage temperature $x$ (in ${}^{\circ} \mathrm { C }$) satisfy the functional relationship $y = e ^ { k x + b }$ ($e = 2.718 \cdots$ is the base of the natural logarithm, $k$ and $b$ are constants). If the shelf life of this food at $0 ^ { \circ } \mathrm{C}$ is designed to be $192$ hours, and the shelf life at $22 ^ { \circ } \mathrm{C}$ is $45$ hours, then the shelf life of this food at $33 ^ { \circ } \mathrm{C}$ is $\_\_\_\_$ hours.

The Logistic model is one of the commonly used mathematical models and can be applied in epidemiology. Based on published data, scholars established a Logistic model for the cumulative confirmed cases of COVID-19 in a certain region $I ( t )$ (where $t$ is measured in days): $I ( t ) = \frac { K } { 1 + \mathrm { e } ^ { -0.23 ( t - 53 ) } }$, where $K$ is the maximum number of confirmed cases. When $I \left( t ^ { * } \right) = 0.95 K$, it indicates that the epidemic has been initially controlled. Then $t ^ { * }$ is approximately (given $\ln 19 \approx 3$)
A. 60
B. 63
C. 66
D. 69

The Logistic model is one of the commonly used mathematical models and can be applied in epidemiology. Based on published data, scholars established a Logistic model for the cumulative confirmed cases $I ( t )$ of COVID-19 in a certain region ($t$ in days): $I ( t ) = \frac { K } { 1 + \mathrm { e } ^ { - 0.23 ( t - 53 ) } }$ , where $K$ is the maximum number of confirmed cases. When $I \left( t ^ { * } \right) = 0.95 K$ , it indicates that the epidemic has been initially controlled. Then $t ^ { * }$ is approximately ( $\ln 19 \approx 3$ )
A. 60
B. 63
C. 66
D. 69

According to experimental statistics, a certain type of bacteria reproduces such that its quantity increases by a factor of 2.4 on average every 3.5 hours. Suppose a test tube in the laboratory initially contains 1000 of this type of bacteria. According to an exponential function model, approximately how many hours later will the quantity of this bacteria reach about $4 \times 10^{10}$? (Note: $\log 2 \approx 0.3010$, $\log 3 \approx 0.4771$)
(1) 63 hours
(2) 70 hours
(3) 77 hours
(4) 84 hours
(5) 91 hours