Question

a bacteria culture initially contains 100 cells and grows at a proportional rate to its size. after an hr the population has increased to 420 a find the expression of bacteria after three hrs b). find the rate of growth after three hrs @Mathematics

Answer 1

let no., at any instant be N.
gn\[dN/dt \alpha N\]
dN/dt=kN........(a)
integrating,\[\int\limits_{?}^{?}dN/N=k \int\limits_{?}^{?}dt\]
lnN=kt+c.........(1)
when ,t=0 & N=100
ln100=c
c=2.....(2)
lnN=kt+2
when , t=1&N=420
ln420=k+2
{2.30300 * log(420) = 6.04134312}
k=4.04.....(3)
[lnN=4.04t+c] is the required eqn.
from (a)
when t=3,
dN/dt=4.04*N.......(b)
but ,lnN=4.04*3+2
2.303*logN=12.12+2
logN=14.12/2.303
logN= 6.1311333
N=1353000........(c)
from (a)&(b)
dN/dt=4.04*1353000
dN/dt=5 466 120per day