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}
d) according to this function, does the medication ever completely leave the bloodstream? explain your answer.
a. yes, as time t increases, the amount of medication in the bloodstream, a(t) also increases.
b. yes, as time t increases, the amount of medication in the bloodstream, a(t) goes to 0.
c. no, since there is a horizontal asymptote at y = 0, there is always some amount of the medication in the bloodstream.
d. no, as time t increases, the amount of medication in the bloodstream, a(t) also increases.

Explanation:

Step1: Analyze the function $A(t)=\frac{A_0}{t^2 + 1}$

We know that $A_0 = 40$ (initial amount). As $t$ increases, the denominator $t^2+1$ increases.

Step2: Determine the behavior of the function as $t$ approaches infinity

We find the limit $\lim_{t
ightarrow\infty}\frac{A_0}{t^2 + 1}$. Since the degree of the denominator is higher than the numerator, $\lim_{t
ightarrow\infty}\frac{40}{t^2 + 1}=0$. But for any finite non - negative $t$, $A(t)>0$.

Step3: Analyze the graph and the long - term behavior

The function $A(t)$ is a rational function. It starts at $A(0) = A_0=40$ (when $t = 0$) and decreases as $t$ increases. It has a horizontal asymptote at $y = 0$.

Answer:

C. No, since there is a horizontal asymptote at $y = 0$, there is always some amount of the medication in the bloodstream.