# Math Help - Population Growth problem

1. ## 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.

2. Hello,
$
\frac{dp}{dt}=k \sqrt{t}
$

Integrate both sides

$
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
-----------------------------------------------------
Try yourself now else see below
so
Now for 7 days
Population
$
=\frac{300*\sqrt{7^3}}{3}+500
$

3. Can you explain what k is in the problem? Is it just a variable for an unknown constant?

4. 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).

5. Thank you adarsh and mr fantastic for the responses.