A bacteria culture initially contains 3000 bacteria and doubles every half hour.
Find the size of the bacterial population after 40 minutes.

Step 1 ：Given that the initial bacteria culture contains 3000 bacteria and doubles every half hour, we are asked to find the size of the bacterial population after 40 minutes.

Step 2 ：This problem is an example of exponential growth, where the population size at any given time is given by the formula \(P = P0 * 2^{\frac{t}{T}}\), where \(P0\) is the initial population size, \(t\) is the time elapsed, and \(T\) is the time it takes for the population to double.

Step 3 ：In this case, \(P0 = 3000\), \(T = 0.5\) hours which is equivalent to 30 minutes, and \(t = 40\) minutes. We need to convert \(t\) and \(T\) to the same units before we can plug them into the formula. Let's convert everything to minutes: \(T = 30\) minutes, \(t = 40\) minutes.

Step 4 ：Now we can calculate \(P\) using the formula \(P = P0 * 2^{\frac{t}{T}}\).

Step 5 ：Substituting the given values into the formula, we get \(P = 3000 * 2^{\frac{40}{30}}\).

Step 6 ：Calculating the above expression, we get \(P \approx 7559.53\).

Step 7 ：Final Answer: The size of the bacterial population after 40 minutes is approximately \(\boxed{7559.53}\).