Step 1 :The problem is asking for an exponential model for the size of the bacteria culture as a function of time. The initial size of the culture is 2000 bacteria and it doubles every 4 hours. This can be represented by the function \(f(t)=2000 \cdot 2^{\left(\frac{t}{4}\right)}\) where \(t\) is the time in hours.
Step 2 :The second part of the question asks to predict the size of the bacteria culture after 2 days. We can substitute \(t\) with the number of hours in 2 days (48 hours) into the function to get the predicted size.
Step 3 :Substitute \(t = 48\) into the function to get the predicted size.
Step 4 :The predicted size of the bacteria culture after 2 days is approximately \(\boxed{8,192,000}\) bacteria.