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:

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<reasoning>

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
\[

$$\begin{aligned} &\text{Degree of numerator } (5t) = 1\\ &\text{Degree of denominator } (3t^2 + 2) = 2\\ &\text{Since } 1 < 2, \lim_{t \to \infty} C(t) = 0\\ &\text{Horizontal asymptote: } y = 0 \end{aligned}$$

\]

Interpret the practical meaning

Using the Rational Function End Behavior knowledge point
\[

$$\begin{aligned} &\lim_{t \to \infty} C(t) = 0 \implies \text{concentration approaches } 0 \text{ over time.}\\ &\text{Meaning: } 0 \text{ is the final amount of the drug left in the bloodstream.} \end{aligned}$$

\]

Match with the correct option

Using the Rational Function End Behavior knowledge point
\[
\text{Option B matches: } y = 0\text{; } 0 \text{ is the final amount, in milligrams, of the drug left.}
\]
</reasoning>

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

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Answer:

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<reasoning>

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
\[

$$\begin{aligned} &\text{Degree of numerator } (5t) = 1\\ &\text{Degree of denominator } (3t^2 + 2) = 2\\ &\text{Since } 1 < 2, \lim_{t \to \infty} C(t) = 0\\ &\text{Horizontal asymptote: } y = 0 \end{aligned}$$

\]

Interpret the practical meaning

Using the Rational Function End Behavior knowledge point
\[

$$\begin{aligned} &\lim_{t \to \infty} C(t) = 0 \implies \text{concentration approaches } 0 \text{ over time.}\\ &\text{Meaning: } 0 \text{ is the final amount of the drug left in the bloodstream.} \end{aligned}$$

\]

Match with the correct option

Using the Rational Function End Behavior knowledge point
\[
\text{Option B matches: } y = 0\text{; } 0 \text{ is the final amount, in milligrams, of the drug left.}
\]
</reasoning>

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

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