Pharmaceutical firms invest significant money in testing any new medication. After the drug is approved for use, it still takes time for physicians to fully accept and start prescribing the medication. The acceptance by physicians approaches a limiting value of $100 \%$, or 1 , after time $\mathrm{t}$, in months. Suppose that the percentage $\mathrm{P}$ of physicians prescribing a new cancer medication is approximated by the equation below. Complete parts (a) through (c).
\[
P(t)=100\left(1-e^{-0.37 t}\right)
\]
a) What percentage of doctors are prescribing the medication after 13 months?
$\%$
(Do not round until the final answer. Then round to the nearest tenth as needed.)
Answer
The percentage of doctors prescribing the medication after 13 months is approximately \(\boxed{99.2\%}\).
Step 1 ：The problem provides the equation \(P(t)=100\left(1-e^{-0.37 t}\right)\) which describes the percentage of physicians prescribing a new cancer medication after time \(t\), in months.
Step 2 ：We are asked to find the percentage of doctors prescribing the medication after 13 months. This can be found by substitifying \(t = 13\) into the given equation.
Step 3 ：Substituting \(t = 13\) into the equation gives \(P(13)=100\left(1-e^{-0.37 \times 13}\right)\).
Step 4 ：Solving this equation gives \(P(13) = 99.185214030232\).
Step 5 ：Rounding to the nearest tenth as required gives the final answer.
Step 6 ：The percentage of doctors prescribing the medication after 13 months is approximately \(\boxed{99.2\%}\).
