Question

An advertising company designs a campaign to introduce a new product to a metropolitan area of population 3 Million people. Let P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 50% of the people were aware of the product after 50 days of advertising. The number of people who become aware of the product at time t is:

Answer 1

`P(t)=3,000,000-3,000,000e^(0.0138t)`

Explanation:

Since P(t) increases at a rate proportional to the number of people still unaware of the product, we have

`P'(t)=K(3,000,000-P(t))`

Since no one was aware of the product at the beginning of the campaign and 50% of the people were aware of the product after 50 days of advertising

P(0) = 0 and P(50) = 1,500,000

We have and ordinary differential equation of first order that we can write

`P'(t)+KP(t)= 3,000,000K`

The integrating factor is

`e^(Kt)`

Multiplying both sides of the equation by the integrating factor

`e^(Kt)P'(t)+e^(Kt)KP(t)= e^(Kt)3,000,000*K`

Hence

`(e^(Kt)P(t))'=3,000,000Ke^(Kt)`

Integrating both sides

`e^(Kt)P(t)=3,000,000K \int e^(Kt)dt +C`

`e^(Kt)P(t)=3,000,000K((e^(Kt))/(K))+C`

`P(t)=3,000,000+Ce^(-Kt)`

But P(0) = 0, so C = -3,000,000

and P(50) = 1,500,000

so

`e^(-50K)=(1)/(2)\Rightarrow K=-(log(0.5))/(50)=0.0138`

And the equation that models the number of people (in millions) who become aware of the product by time t is

`P(t)=3,000,000-3,000,000e^(0.0138t)`