a mathematical consultant determines that the proportion of people who will have responded to the advertisement of a new product after it has been marketed for t days is given by f(t)=0.9(1-e^-0.3t).
a) if the add in on the air for 60 days, when will the maximum proportion of people have responded? (check endpoints and critical points)
b) how can you determine the answer without doing any calculations simply by looking at the equation.
f(t)=0.9(1-e^-0.3t) = .9 - .9e^(-.03t)
f ' (t) = -.9(-.3)(e^(-.03t) ) = 0 for a max/min
e^(-.03t) = 0 ---> no solution, so let's look at endpoints
f(1) = .9 - .9e^0 = .9 - .9 = 0 , that makes sense
f(60) = .9 - .9e^(-1.8)
= .9 - .1488
= .72123 or appr 72%
b) sure I can. Look at the e^(-.03t).
As t gets larger, e^(-.03t) gets smaller and smaller
so you end up with .9 - appr. zero
or just .9 which is 90%
So in the long run, 90% of the people will eventually respond to the ad