Question

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 t, in months. suppose that the percentage p of physicians prescribing a new cancer medication is approximated by the equation below. complete parts (a) through (c) p(t)=100(1 - e^(-0.43t))
a) what percentage of doctors are prescribing the medication after 13 months?
99.6 %
(do not round until the final answer. then round to the nearest tenth as needed.)
b) find p(13), and interpret its meaning.
p(13)=
(do not round until the final answer. then round to the nearest tenth as needed )

Explanation:

Step1: Find the derivative of $P(t)$

We use the chain - rule. If $P(t)=100(1 - e^{-0.43t})$, then $P'(t)=100\times0.43e^{-0.43t}=43e^{-0.43t}$.

Step2: Evaluate $P'(13)$

Substitute $t = 13$ into $P'(t)$. So $P'(13)=43e^{-0.43\times13}$.
First, calculate the exponent: $-0.43\times13=-5.59$.
Then, find $e^{-5.59}\approx0.0037$.
Multiply by 43: $P'(13)=43\times0.0037 = 0.1591\approx0.2$.

The meaning of $P'(13)$ is that 13 months after the drug is approved, the rate of change of the percentage of physicians prescribing the new cancer medication is approximately $0.2$ percentage points per month.

Answer:

$0.2$