Question

a drug is injected into a patient and the concentration of the drug is monitored. the drugs concentration, \\(c(t)\\), in milligrams after \\(t\\) hours is modeled by \\(c(t) = \frac{5t}{3t^2 + 2}\\). what is the horizontal asymptote for this function? describe what this means in practical terms.

a. \\(y = 1.67\\); 1.67 is the final amount, in milligrams, of the drug that will be left in the patients bloodstream.
b. \\(y = 0\\); 0 is the final amount, in milligrams, of the drug that will be left in the patients bloodstream.
c. \\(y = 1.67\\); after 1.67 hours, the concentration of the drug is at its greatest.
d. \\(y = 1.00\\); after 1.00 hours, the concentration of the drug is at its greatest.

Explanation:

Identify the given function

Using the Rational Functions knowledge point

$$ C(t) = \frac{5t}{3t^2 + 2} $$

Determine the horizontal asymptote

Using the Horizontal Asymptotes and Rational Function End Behavior knowledge points

$$
LATEXBLOCK0
$$

Interpret the practical meaning

Using the Rational Function End Behavior knowledge point

$$
LATEXBLOCK1
$$

Match with the correct option

Using the Rational Function End Behavior knowledge point

$$ \text{Option B matches: } y = 0\text{; } 0 \text{ is the final amount, in milligrams, of the drug left.} $$

Answer:

• A. \(y = 1.67\); 1.67 is the final amount, in milligrams, of the drug that will be left in the patient's bloodstream.
• B. \(y = 0\); 0 is the final amount, in milligrams, of the drug that will be left in the patient's bloodstream. (Correct answer)
• C. \(y = 1.67\); After 1.67 hours, the concentration of the drug is at its greatest.
• D. \(y = 1.00\); After 1.00 hours, the concentration of the drug is at its greatest.