Problem

The number of milligrams $D(h)$ of a drug in a patient's bloodstream $h$ hours after the drug is injected. is modeled by the following function. \[ D(h)=50 e^{-0.2 h} \] Find the initial amount injected and the amount in the bloodstream after 7 hours. Round your answers to the nearest hundredth as necessary.

Solution

Step 1 :The initial amount of the drug injected can be found by evaluating the function at \(h=0\).

Step 2 :Substitute \(h=0\) into the function \(D(h)=50 e^{-0.2 h}\) to get \(D(0)=50 e^{-0.2*0}=50.0\).

Step 3 :The amount of the drug in the bloodstream after 7 hours can be found by evaluating the function at \(h=7\).

Step 4 :Substitute \(h=7\) into the function \(D(h)=50 e^{-0.2 h}\) to get \(D(7)=50 e^{-0.2*7}\) which is approximately 12.33.

Step 5 :Final Answer: The initial amount of the drug injected is \(\boxed{50.0}\) milligrams and the amount in the bloodstream after 7 hours is approximately \(\boxed{12.33}\) milligrams.

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Source: https://solvelyapp.com/problems/20438/

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