Evaluating an exponential function with base e that models a real-world.
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 functio
\[
D(h)=45 e^{-0.3 h}
\]
Find the initial amount injected and the amount in the bloodstream after 6 hours.
Round your answers to the nearest hundredth as necessary.
Initial amount:
milligrams
Amount after 6 hours:
milligrams
Final Answer: The initial amount of the drug injected is \(\boxed{45.00}\) milligrams and the amount in the bloodstream after 6 hours is approximately \(\boxed{7.44}\) milligrams.
Step 1 :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 function $D(h)=45 e^{-0.3 h}$.
Step 2 :The initial amount of the drug can be found by evaluating the function at $h=0$.
Step 3 :The amount of the drug after 6 hours can be found by evaluating the function at $h=6$.
Step 4 :Calculating these values, we find that the initial amount is 45.0 milligrams and the amount after 6 hours is approximately 7.44 milligrams.
Step 5 :Final Answer: The initial amount of the drug injected is \(\boxed{45.00}\) milligrams and the amount in the bloodstream after 6 hours is approximately \(\boxed{7.44}\) milligrams.