Applic cal to phy world expo (1 Viewer)

[chewy123]

A population in a certain city is growing at a rate proportional to the population itself. After 3 yrs the population increases by 20%. How long will it take for the population to double.

thx

 

[vds700]

A population in a certain city is growing at a rate proportional to the population itself. After 3 yrs the population increases by 20%. How long will it take for the population to double.

thx

dP/dt = kP
dt/dP = 1/kP
t = (1/k) . Int (1/P)dP
=(1/k)lnP + c
when t = 0, P = P₀ (P₀ is the initial population, when t = 0)
0 = (1/k)ln P₀ + c
therefore c = -(1/k)ln P₀
t= (1/k)lnP - (1/k)ln P₀
=(1/k)ln(P/ P₀ )
when t = 3. P = 1.2 P₀
3 = (1/k)ln(1.2)
k = (1/3)ln(1.2) = 0.06077
therefore t = (1/0.06077)ln(P/ P₀)
when P = 2 P₀ ,
t = (1/0.06077)ln(2)
=11.41 years

 

[3unitz]

little different working to vds700:

N = N₀e^kt

N₀e^k3 / N₀e^k0 x 100 = 120 (population increases by 20%)

e^k3 = 12/10

k3 = ln(12/10)

k = (1/3) ln(12/10)

let T denote time it takes for population to double:

N₀e^kT / N₀e^k0 = 2 (population doubles)

e^kT = 2

kT = ln 2

T = ln2 / k

T = ln2 / [(1/3) ln(12/10)]

T ~ 11.41 years