# Homework Help: Problem with ODE

1. Feb 17, 2006

Hi please can I have some help with this problem:

An advertising company designs a campaign to introduce a new product to a metropolitan area of population 4 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 10% of the people were aware of the product after 30 days of advertising. What is the number of people who become aware of the product at time t .

I can not set the equation...

Thank you

B

2. Feb 17, 2006

### dicerandom

There are a few key phrases here.

"Suppose that P increases at a rate proportional to the number of people still unaware of the product."

This gives you a relationship between P(t) and its time derivative.

"The company determines that no one was aware of the product at the beginning of the campaign, and that 10% of the people were aware of the product after 30 days of advertising."

This gives you your boundary conditions.

Give it a try and post back here if you're still having troubles.

3. Feb 17, 2006

### benorin

This is a logistic growth problem, similar to exponential growth problems only with the further knowledge of finite resources, e.g. 4 million people.

4. Feb 18, 2006

### HallsofIvy

If P(t) is the number of people who are aware of the product and there are a total of 4000000 people, how many are not aware of the product? "Proportional to" means, of course, a constant times that.

This not, precisely, a "logistic" equation.