# problem with ODE

• Feb 17th 2006, 06:15 PM
problem with ODE
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 up the equation...

My problem I think is thie sentence:
Suppose that P increases at a rate proportional to the number of people still unaware of the product
I found something like
dP/dt= k * ( Po- P(t) ) where Po is the original population. But I am not sure

Thank you

B
• Feb 17th 2006, 09:19 PM
CaptainBlack
Quote:

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 up the equation...

My problem I think is thie sentence:
Suppose that P increases at a rate proportional to the number of people still unaware of the product
I found something like
dP/dt= k * ( Po- P(t) ) where Po is the original population. But I am not sure

Thank you

B

Looks OK to me, that is exactly what the problem says, and is one of
the common models for this sort of problem.

It essentially means some thing like:

Everyone not aware of the product has an equal chance of becoming aware
of it in unit time, so the infection rate is proportional to the number not
infected. Then we treat the population as though its a continuous variable
rather than discrete, which give us your ODE:

$\displaystyle \frac{dP}{dt}=k(P_0-P)$,

with initial condition $\displaystyle P(0)=0$.

RonL
• Feb 18th 2006, 12:25 AM
ticbol
Quote:

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 up the equation...

My problem I think is thie sentence:
Suppose that P increases at a rate proportional to the number of people still unaware of the product
I found something like
dP/dt= k * ( Po- P(t) ) where Po is the original population. But I am not sure

Thank you

B

Here is one way.

If P(t) = number of people who become aware after time t, then,
[4 -P(t)] = number of people still unaware.
So,
dP(t) /dt = k[4 -P(t)] -----------(i)
where
---P(t) is read "P of t", or "P as a function of t", in Millions of persons.
---t is time, in number of days.
---k is constant of proportionality.

For less confusion, let P = P(t), then (i) becomes
dP/dt = k*(4-P) ---------------------------------------(ia)
dP = k*(4-P)*dt
dP/(4-P) = k*dt
Integrate both sides,
-ln(4-P) = k*t +C ---------------(ii)

The problem says:
---When t=0, P(0) = none = 0 also. ----------------------(1)
---when t=30days, P(30) = 10% of 4Million = 0.4M --------(2)

Apply condition (1) into (ii),
-ln(4-0) = k*0 +C
So, C = -ln(4) -----------------***

Apply that, and condition (2), into (ii),
-ln(4 -0.4) = k*30 -ln(4)
Simplifying,
ln(4) -ln(3.6) = 30k
ln[4/3.6] = 30k
k = (1/30)ln(1/0.9) = (1/30)[ln(1) -ln(0.9)]
k = -(1/30)ln(0.9)-----------------------------------***

To find P in general, going back to (ii),
-ln(4-P) = kt -ln(4)
ln(4) -ln(4-P) = kt
ln[4/(4-P)] = kt
4/(4-P) = e^(kt)
The reciprocal of the LHS = the reciprocal of the RHS,
(4-P)/4 = e^(-kt)
4-P = 4e^(-kt)
P = 4 -4e^(-kt)

where k = -(1/30)ln(0.9), so,
P = 4[1 -e^(ln(0.9) *t/30)] --------------answer.

If we go back to P as function of t, or P(t), and let Po = total population, then,
Etc.
• Feb 18th 2006, 01:41 PM
Quote:

Originally Posted by ticbol
Here is one way.

If P(t) = number of people who become aware after time t, then,
[4 -P(t)] = number of people still unaware.
So,
dP(t) /dt = k[4 -P(t)] -----------(i)
where
---P(t) is read "P of t", or "P as a function of t", in Millions of persons.
---t is time, in number of days.
---k is constant of proportionality.

For less confusion, let P = P(t), then (i) becomes
dP/dt = k*(4-P) ---------------------------------------(ia)
dP = k*(4-P)*dt
dP/(4-P) = k*dt
Integrate both sides,
-ln(4-P) = k*t +C ---------------(ii)

The problem says:
---When t=0, P(0) = none = 0 also. ----------------------(1)
---when t=30days, P(30) = 10% of 4Million = 0.4M --------(2)

Apply condition (1) into (ii),
-ln(4-0) = k*0 +C
So, C = -ln(4) -----------------***

Apply that, and condition (2), into (ii),
-ln(4 -0.4) = k*30 -ln(4)
Simplifying,
ln(4) -ln(3.6) = 30k
ln[4/3.6] = 30k
k = (1/30)ln(1/0.9) = (1/30)[ln(1) -ln(0.9)]
k = -(1/30)ln(0.9)-----------------------------------***

To find P in general, going back to (ii),
-ln(4-P) = kt -ln(4)
ln(4) -ln(4-P) = kt
ln[4/(4-P)] = kt
4/(4-P) = e^(kt)
The reciprocal of the LHS = the reciprocal of the RHS,
(4-P)/4 = e^(-kt)
4-P = 4e^(-kt)
P = 4 -4e^(-kt)