Chapter 3.7, Problem 84E

Drug Concentration After a certain drug is injected into a patient, the concentration c of the drug in the bloodstream is monitored. At time t ≥ 0 (in minutes since the injection) the concentration (in mg/L) is given by

$c(t)=\frac{30t}{t^2+2}$

1. (a) Draw a graph of the drug concentration.
2. (b) What eventually happens to the concentration of drug in the bloodstream?

To determine

To sketch: The graph of the drug concentration in the bloodstream $p\left(t\right)=\frac{30t}{t^2+1}$.

Explanation of Solution

The drug concentration in the bloodstream is given below, $p\left(t\right)=\frac{30t}{t^2+1}$ where t is the time in minutes since the injection and c is the concentration of drug in bloodstream.

$c\left(t\right)$ is a given rational function.

The t-intercepts are the zeros of the numerator of rational function, $t=0$.

So, the t-intercept of rational function is 0.

Substitute 0 for t in rational function to find the y-intercept of the function,

$\begin{array}{c}p\left(0\right)=\frac{30\left(0\right)}{0^2+1}\\ =\frac{0}{1}\\ =0\end{array}$

So, the y- intercept of given rational function is 0.

There are two asymptotes: Vertical asymptote and Horizontal asymptote.

The vertical asymptotes occur where the denominator is zero and where the function is undefined.

The zeros of denominator of rational function,

$\begin{array}{l}{t}^2=-2\\ t=\pm \sqrt{-2}\end{array}$

This is not a real value, so, the vertical asymptotes do not exist.

The highest power of numerator is less than the highest power of dominator, so, the horizontal asymptote is $y=0$.

Substitute different values of t to sketch the graph of function $p\left(t\right)=\frac{30t}{t^2+1}$

$t$	$y$
$1$	10
$2$	10
$-2$	-10
$-3$	-8.1

Sketch the graph of given rational function by given above all information,

Figure (1)

Figure (1) shows the graph of drug concentration.

To determine

The drug concentration through graph.

Explanation of Solution

The graph of rabbit population $p\left(t\right)=\frac{30t}{t^2+1}$ is given in Figure (1).

From Figure (1), identify that the drug concentration increases at 10.