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))
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)=0.2
(do not round until the final answer. then round to the nearest tenth as needed.)
choose the correct interpretation below
a. at 13 months, the percentage of doctors who are prescribing the medication is increasing by 0.2% per month
b. at 13 months, 0.2% of doctors are not prescribing the medication
c. at 13 months, 0.2% of doctors are prescribing the medication
d. at 13 months, the percentage of doctors who are prescribing the medication is decreasing by 0.2% per month

Explanation:

Step1: Recall derivative rules

The derivative of a constant - multiple function \(y = a\cdot f(x)\) is \(y'=a\cdot f'(x)\), and the derivative of \(e^{-ax}\) is \(-a\cdot e^{-ax}\). Given \(P(t)=100(1 - e^{-0.43t})=100-100e^{-0.43t}\). The derivative \(P'(t)\) of \(P(t)\) is \(P'(t)=100\times0.43e^{-0.43t}=43e^{-0.43t}\).

Step2: Evaluate \(P'(13)\)

Substitute \(t = 13\) into \(P'(t)\): \(P'(13)=43e^{-0.43\times13}=43e^{-5.59}\). Calculate \(e^{-5.59}\approx0.00395\), then \(P'(13)=43\times0.00395\approx0.2\).

Step3: Interpret the derivative

The derivative \(P'(t)\) represents the rate of change of the percentage of physicians prescribing the new - cancer medication with respect to time \(t\) (in months). A positive value of \(P'(t)\) means the percentage is increasing. When \(t = 13\), \(P'(13)\approx0.2\), which means at 13 months, the percentage of doctors who are prescribing the medication is increasing by \(0.2\%\) per month.

Answer:

A. At 13 months, the percentage of doctors who are prescribing the medication is increasing by 0.2% per month.