Question

a drug is injected into a patient and the concentration of the drug in the bloodstream is monitored. the drug’s concentration, c(t), in milligrams per liter, after t hours is modeled by the equation below. the graph of this rational function is shown to the right.
$c(t) = \frac{4t}{t^2 + 2}$
complete parts a through c below.

a. use the graph to obtain a reasonable estimate of the drug’s concentration after 2 hours.
the drug’s concentration after 2 hours is approximately 1.3 milligrams per liter.
(round to the nearest tenth as needed.)

b. use the function’s equation to determine the drug’s concentration after 2 hours.
the drug’s concentration after 2 hours is 1.3 milligrams per liter.
(round to the nearest tenth as needed.)

c. use the function’s equation, $c(t) = \frac{4t}{t^2 + 2}$, to find the horizontal asymptote for the graph. describe what this means about the drug’s concentration in the patient’s bloodstream as time increases.
select the correct choice below and, if necessary, fill in the answer box to complete your answer.

○ a. the horizontal asymptote is
(type an equation.)

○ b. there is no horizontal asymptote.

Explanation:

Step1: Determine the degrees of numerator and denominator

For the function \(C(t)=\frac{4t}{t^{2}+2}\), the degree of the numerator \(n = 1\) (since the highest - power of \(t\) in the numerator is \(t^1\)) and the degree of the denominator \(m=2\) (since the highest - power of \(t\) in the denominator is \(t^2\)).

Step2: Apply the rule for horizontal asymptotes of rational functions

The rule for horizontal asymptotes of a rational function \(y = \frac{f(t)}{g(t)}\) where \(f(t)=a_{n}t^{n}+\cdots\) and \(g(t)=b_{m}t^{m}+\cdots\) is as follows:
If \(n\lt m\), then \(y = 0\) is the horizontal asymptote.

Since \(n = 1\) and \(m = 2\) (\(1\lt2\)), the horizontal asymptote is \(y = 0\).

In the context of the drug - concentration problem, as \(t\) (time) gets larger and larger (i.e., as \(t
ightarrow\infty\)), the value of \(C(t)=\frac{4t}{t^{2}+2}\) approaches \(0\). This means that as time increases, the concentration of the drug in the patient's bloodstream approaches \(0\) milligrams per liter.

Answer:

A. The horizontal asymptote is \(y = 0\)