Population Grows

Question

A population grows exponentially according to the differential equation dp/dt =kp, where P is the population, t is time, and k is a positive constant. If P(0) = A, what is the time for the population to quadruple its initial value? 

4A

ln(k)/4

k/ln4

ln4/k

 

Answer 1

Separate the variables

 

dp/p=kdt

 

Integrate both sides. 

 

∫dp/p=∫k dt

 

ln P = kt +C

 

Take an e of both sides to get rid of the ln

 

You will get

 

P=e^kt+e^C

 

You an combine this and get

 

P=Ce^kt

 

Now they have given you some information.  At time 0 your population is A.  You will plug this in. 

 

A = Ce^0t Anything to the 0th power is = 1.  Therefore C = A

 

Now it is asking for the time it requires for the population to quadruple. 

 

You will plug it in like this:

 

4A=Ae^kt

 

Divide 4A by A and get

 

4=e^kt. Take ln of both sides

 

ln4=kt then divide by k to solve for t. 

 

Therefore t=ln4/k