Questions: 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)=40 e^(-0.3 t). How many milligrams of the drug remain in the patient's bloodstream after 5 hours? (Round your answer to one decimal place.) 5.41

Solution Steps

To determine the number of milligrams of the drug remaining in the patient's bloodstream after 5 hours, we need to evaluate the given function \( D(t) = 40 e^{-0.3t} \) at \( t = 5 \). This involves substituting \( t = 5 \) into the function and calculating the result.

We start with the function that models the amount of the drug remaining in the bloodstream after \( t \) hours, given by

\[ D(t) = 40 e^{-0.3t}. \]

To find the amount remaining after \( t = 5 \) hours, we substitute \( t \) with 5:

\[ D(5) = 40 e^{-0.3 \cdot 5}. \]

Next, we calculate the exponential term:

\[ -0.3 \cdot 5 = -1.5. \]

Thus, we have:

\[ D(5) = 40 e^{-1.5}. \]

Now we compute \( e^{-1.5} \):

\[ e^{-1.5} \approx 0.22313. \]

Substituting this value back into the equation gives:

\[ D(5) \approx 40 \cdot 0.22313 \approx 8.9252. \]

Rounding this to one decimal place, we find:

\[ D(5) \approx 8.9. \]

Final Answer