An advertising company designs a campaign to introduce a newproduct to a metropo

Question

An advertising company designs a campaign to introduce a newproduct to a metropolitan area of population 4 Million people. LetP(t) denote the number of people (in millions)who become aware of the product by time t. Suppose thatP increases at a rate proportional to the number of peoplestill unaware of the product. The company determines that no onewas aware of the product at the beginning of the campaign, and that10% of the people were aware of the product after 30 days ofadvertising. The number of people who become aware of the productat time t is:

Explanation / Answer

Let N equal total population (4,000,000) This can be treated as a calculus problem The number of people unaware of the product is N - P(t) Since the rate of change of P(t) is proportional to the number ofpeople still unaware: d P(t)/dt = k( N - P(t)) , where k is some unknown constant Now d [N-P(t)]/dt = - d P(t)/dt so: - d[N - P(t)]/[N - P(t)] = k dt Integrating both sides: log [ N - P(t)] = -kt + c , where c is an unknown constant ofintegration Now equate e^{log [N - P(t)]} to e^{-kt + c} N - P(t) = e^{-kt + c} When t = 0, P(t) = 0 implies N = e^c so that c = log N Therefore P (t) = N - e^{-kt + log N} or P (t) = N (1 -e^{-kt}) The condition that 10 % of the people are aware of the productafter 30 days implies that: P (30)/N = 0.1 implies e^{-30k} = 0.9 or - 30 k = log 0.9 and k = log 0.9/(- 30) = .003512 Finally P (t) = N ( 1 - e^{-.003512 t}), where N = 4,000,000 inthis case.

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