Solve the problem.
-A drug is injected into a patient and the concentration of the drug is monitored. The drug's concentration, $C ( t )$, in milligrams after $t$ hours is modeled by
$$C ( t ) = \frac { 5 t } { 3 t ^ { 2 } + 1 } \text {. }$$
What is the horizontal asymptote for this function? Describe what this means in practical terms.
A) $\mathrm { y } = 0 ; 0$ is the final amount, in milligrams, of the drug that will be left in the patient's bloodstream.
B) $\mathrm { y } = 1.67 ; 1.67$ is the final amount, in milligrams, of the drug that will be left in the patient's bloodstream.
C) $\mathrm { y } = 1.25$; After $1.25$ hours, the concentration of the drug is at its greatest.
D) $\mathrm { y } = 1.67$; After $1.67$ hours, the concentration of the drug is at its greatest.

Question

Quizplus · Accepted Answer

To find the horizontal asymptote of the function $C(t) = \frac{5t}{3t^2 + 1}$, we look at the behavior as $t$ approaches infinity. The degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, which means the horizontal asymptote is $y = 0$. In practical terms, this means that over time, the concentration of the drug in the patient's bloodstream approaches zero, indicating that eventually, the drug will be fully metabolized or eliminated from the body.

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