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)

P = 4[1 -e^(-kt)] ----------------------answer.

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,

P(t) = (Po)[1 -e^(-kt)] ------------answer.

Etc.