# Integration (population growth)

• Jan 11th 2009, 09:54 AM
saiyanmx89
Integration (population growth)
The rate of growth dP/dt of a population of bacteria is proportional to the square root of t, where P is the population size and t is the time in days (t= [0,10]).

dP/dt= k*sqrt(t)

The initial size of the population is 500. After 1 day the population has grown to 600. Estimate the population after 7 days.
• Jan 11th 2009, 10:56 AM
vincisonfire
$*\frac{dP}{dt} = k\sqrt{t}$*
$\int dP = \int k\sqrt{t}dt$
$P(t) = \frac{2k}{3}t^{\frac{3}{2}} + C$
$P(0) = 500 = C$
$P(1) = 600 = \frac{2k}{3} + 500 \implies k = 150$
$P(t) = 100t^{\frac{3}{2}} + 500$