$$Question: A pharmacologist models the concentration of a drug in the bloodstream with the function $ h(t) = t^2 - 4t + 5m $, where $ t $ is time in hours and $ m $ is a dosage parameter. If at $ t = 3 $, the concentration equals $ 10 $, what is the value of $ m $?

Understanding Drug Concentration: Solving for the Dosage Parameter $ m $

In pharmacokinetics, modeling drug concentration over time is essential for determining effective and safe dosages. A recent pharmacological study uses the function $ h(t) = t^2 - 4t + 5m $ to represent the concentration of a drug in the bloodstream $ t $ hours after administration. Given that at $ t = 3 $, the concentration $ h(t) $ equals 10, this article explains how to solve for the dosage parameter $ m $.

Setting Up the Equation

Understanding the Context

We are given:
$$
h(t) = t^2 - 4t + 5m
$$
and the condition:
$$
h(3) = 10
$$

Substitute $ t = 3 $ into the function:
$$
h(3) = (3)^2 - 4(3) + 5m = 10
$$
$$
9 - 12 + 5m = 10
$$
$$
-3 + 5m = 10
$$

Solving for $ m $

Add 3 to both sides:
$$
5m = 13
$$

$$Question: A pharmacologist models the concentration of a drug in the bloodstream with the function $ h(t) = t^2 - 4t + 5m $, where $ t $ is time in hours and $ m $ is a dosage parameter. If at $ t = 3 $, the concentration equals $ 10 $, what is the value of $ m $?

Key Insights

Divide both sides by 5:
$$
m = rac{13}{5}
$$

Thus, the dosage parameter $ m $ is $ rac{13}{5} $.

Why This Matters in Pharmacology

Accurate modeling of drug concentration helps clinicians optimize dosing schedules and maintain therapeutic levels without toxicity. By plugging in real-world measurements (like blood concentration at a specific time), pharmacologists use equations like $ h(t) $ to fine-tune dosage parameters—ensuring patient safety and treatment efficacy.

Final Answer

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Final Thoughts

$$
oxed{rac{13}{5}}
$$

Understanding this model and solving for $ m $ enhances precision in drug therapy and reflects the vital role of mathematical modeling in modern medicine.