Problem

After three hours, we observe the bacteria colony and find the population is now 1600. Use this information to solve the differential equation in the previous example, and find a more accurate constant of proportionality. How does this model differ from the previous one? Again, predict the population at t = 6 hours and compare with the previous estimate.

Solution

The differential equation has the form

Using separation of variables, we find the solution is

Since we have two known points on this curve (t = 0, P = 100 and t = 3, P = 1600) we can solve for the two constants k and C.

So the equation is

and when t = 6,

This solution, still exponential, differs from the previous solution because the constant k is smaller, resulting in slower growth than the first guess at the solution.