Problem Statement no.2 A useful new product is introduced into an isolated fixed population of 1,000,000 people, and 100 of these people adopt this product initially, that is, at time t= 0. Suppose the rate at which the product is adopted is proportional to the number of the people who have adopted it already multiplied by the number of them who have not yet done so. If we let x denote the number of people who have adopted the product at time t, measured in then we have the initial-value problem dx dt = kx(1,000,000 - x) x(0) = 100, where k is the constant of proportionality. (a) Solve this initial-value problem (b) How many people have adopted the product after two weeks? (c) When will one half of the given population have adopted it? Solution:
Problem Statement no.2 A useful new product is introduced into an isolated fixed population of 1,000,000 people, and 100 of these people adopt this product initially, that is, at time t= 0. Suppose the rate at which the product is adopted is proportional to the number of the people who have adopted it already multiplied by the number of them who have not yet done so. If we let x denote the number of people who have adopted the product at time t, measured in then we have the initial-value problem dx dt = kx(1,000,000 - x) x(0) = 100, where k is the constant of proportionality. (a) Solve this initial-value problem (b) How many people have adopted the product after two weeks? (c) When will one half of the given population have adopted it? Solution:
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