Medical Drugs When a certain medical drug is administered to...

Medical Drugs When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient's bloodstream after $t$ hours is modeled by
$$D(t)=50 e^{-0.2 t}$$
How many milligrams of the drug remain in the patient's bloodstream after 3 hours?

Key Concepts

Exponential Decay

Exponential decay is a process in which a quantity decreases at a rate proportional to its current value. This is typically represented by equations of the form A(t) = A?e^(–kt), where A? is the initial amount, k is a positive constant that determines the decay rate, and t is time. This concept is essential for modeling situations where substances diminish over time, such as medicine in the bloodstream, radioactive materials, or cooling objects.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine the corresponding output. In the context of mathematical modeling, this means plugging in a particular time into an exponential decay function to calculate the remaining amount of a substance. This process is fundamental in applying mathematical models to real-world scenarios, ensuring accurate predictions based on given input parameters.

Modeling with Exponential Functions

Modeling with exponential functions is a critical technique used in various fields to represent processes involving rapid change over time. These models help scientists and practitioners predict outcomes such as drug concentration levels in the human body, population growth, or decay phenomena. Understanding how to interpret and evaluate these functions is vital for making informed decisions and accurate calculations in practical applications.

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