## **Core Concept**
The problem involves pharmacokinetics, specifically the calculation of the dose of a drug administered intravenously (IV) based on its volume of distribution (Vd), elimination half-life, and plasma concentration at a given time. The formula to calculate the concentration of a drug at any time point after IV administration is (C(t) = frac{Dose}{Vd} times e^{-lambda t}), where (C(t)) is the concentration at time (t), (Dose) is the administered dose, (Vd) is the volume of distribution, (lambda) is the elimination rate constant ((lambda = frac{0.693}{t_{1/2}})), and (t) is the time.

## **Why the Correct Answer is Right**
Given that the plasma level of the drug at 12 hours ((t = 12) h) is 3 mg/L, (Vd = 10) L, and the elimination half-life ((t_{1/2})) = 6 h, we can calculate the dose. First, we find (lambda): (lambda = frac{0.693}{6} = 0.1155) h(^{-1}). Rearranging the formula to solve for (Dose), we get (Dose = C(t) times Vd times e^{lambda t}). Substituting the given values: (Dose = 3 times 10 times e^{0.1155 times 12}). Calculating (e^{0.1155 times 12} = e^{1.386} approx 4). So, (Dose = 3 times 10 times 4 = 120) mg.

## **Why Each Wrong Option is Incorrect**
- **Option A:** Without calculation, it's hard to directly refute, but let's assume it's less than 120 mg based on the provided correct answer.
- **Option B:** Similarly, assume it's incorrect based on the correct answer provided.
- **Option D:** This option suggests a dose of 30 mg, which seems too low given the parameters (e.g., (Vd = 10) L, concentration at 12 h is 3 mg/L).

## **Clinical Pearl / High-Yield Fact**
A key concept here is understanding the relationship between a drug's volume of distribution, half-life, and plasma concentration over time. The volume of distribution ((Vd)) helps in estimating the dose required to achieve a desired plasma concentration. This calculation is crucial in clinical pharmacology for ensuring efficacy and safety of drugs.

## **Correct Answer:** . 120 mg