Q:A bacteria culture initially contains 2500 bacteria and doubles every half hour. What is the size of the baterial population after 20 minutes?

A:

Known: Ao = initial bacterial quantity of 2500; and the bacteria quantity is doubling every half hour.

Unknown: We need to find the amount of bacteria after 20 minutes of growth.

To do this we must first find a function of the bacteria quantity for a time: A(t).

The first step is to express the relationship between quantity and rate of change measured in bacteria per hour as a differential equation:

dA/dt = kA. where A is the quantity of bacteria at a given time and k is the relative growth rate.

The formula for finding the relative growth rate is:

(ln(y2)-ln(y1))/(x2-x1)->(ln(2Ao)-ln(Ao))/(1/2)=2ln(2)= 1.386

This is a seperable equation so: dA/A=kdt. Integrating both sides yields -> ln (A) = kt + C where C is a constant.

Taking the exponential of both sides gives us: A = Ce^kt.

We have a function for the quantity of bacteria but we must find the constant. We can find the constant by using the initial conditions at time zero.

2500 = (C*e^0) -> C=2500=Ao.

A = Ao*e^kt where Ao = 2500 bacteria and k = 1.386

So plugging in 20 minutes or 1/3 hour into the function gives us:

A=2500*e^(1.386*1/3)= 3968 bacteria.