Question

after an injection, the amount of a medication a in cubic centimeters (cc) in the bloodstream decreases with time t in hours. suppose that under certain conditions a is given by the function below, where a₀ is the initial amount of the medication. assume that an initial amount of 40 cc is injected. complete parts (a) through (d)
a(t)=\frac{a₀}{t² + 1}
(type an integer or decimal rounded to the nearest ten - thousandth as needed.)
a(10)=0.3960 cc
(type an integer or decimal rounded to the nearest ten - thousandth as needed.)
b) find the maximum amount of medication in the bloodstream over the interval 0,∞)
the maximum amount of medication is 40 cc at t = 0
c) sketch a graph of the function
choose the correct graph below

Explanation:

Step1: Analyze the function behavior

We have the function $A(t)=\frac{A_0}{t^{2}+1}$, with $A_0 = 40$. As $t$ increases from $0$ to $\infty$, the denominator $t^{2}+1$ increases. Since the numerator is constant ($A_0 = 40$), the value of the function $A(t)$ decreases. At $t = 0$, $A(0)=\frac{40}{0^{2}+1}=40$. As $t
ightarrow\infty$, $\lim_{t
ightarrow\infty}\frac{40}{t^{2}+1}=0$.

Step2: Match with graph

The function starts at $A(0) = 40$ and decreases as $t$ increases. Graph - C shows a function that starts at a non - zero value on the $y$ - axis and decreases as $x$ (in this case $t$) increases.

Answer:

C