# Population Growth problem

• Jan 8th 2009, 08:45 PM
yeloc
Population Growth problem
The rate of growth of 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.

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 8th 2009, 09:30 PM
Hello(Hi),
$\displaystyle \frac{dp}{dt}=k \sqrt{t}$
Integrate both sides

$\displaystyle p=\frac{2k*\sqrt{t^3}}{3}+c$

for t=0 p=500
hence c=500

for t=1
p=2k/3+c=600
thus
k=150
-----------------------------------------------------
(Wait)Try yourself now else see below
so
Now for 7 days
Population
$\displaystyle =\frac{300*\sqrt{7^3}}{3}+500$
• Jan 8th 2009, 09:37 PM
yeloc
Can you explain what k is in the problem? Is it just a variable for an unknown constant?
• Jan 8th 2009, 09:41 PM
mr fantastic
Quote:

Originally Posted by yeloc
Can you explain what k is in the problem? Is it just a variable for an unknown constant?

k represents a constant whose value you have to find (actually I should use past tense since the value has been found).
• Jan 8th 2009, 09:43 PM
yeloc
Thank you adarsh and mr fantastic for the responses.