Problem

More research indicates that the growth of the bacteria colony is proportionally dependent on the amount of oxygen available. In the controlled environment where our sample resides, all the oxygen will be exhausted in 3 days (72 hours). Thus, our new differential equation modeling the bacteria growth is

Determine the population at time t = 6 hours, and describe the long term behavior of the bacteria colony.

Solution

We use separation of variables to solve the latest refinement of the differential equation for P in terms of t.

The initial condition t = 0, P = 100 gives us C = 100, as before. To solve for k, we use our understanding that dP/dt is positive for t < 72, negative for t > 72, and 0 when t = 72. We differentiate the formula for P, set it equal to 0, let t = 72 and solve for k.

Since the exponential factor in the equation is never 0, this has solutions k = 0 and k = 1. k = 0 does not make sense, since there would be no growth at all, so we have

The new estimate at t = 6 is

With the previous equations we used to model the behavior of the bacteria colony, the population would increase exponentially, forever. The new equation is more realistic in that this does not happen. As the time t approaches 72 hours, the rate of growth dP/dt slows down—the resources are being used up. At t = 72 hours, the oxygen is used up and growth ceases, as the population is at its peak. As time continues, dP/dt becomes negative, indicating that the bacteria have begun to die off. Finally, after about 148.4 hours, the population has dropped to about 1, and it too soon dies off.

accurate to 3 decimal places