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 \(\boxed{99.2\%}\).