The count in a bacteria culture was 900 after 20 minutes and 1400 after 35 minutes. Assuming the count grows exponentially,
What was the initial size of the culture?
Find the doubling period.
Find the population after 85 minutes.
When will the population reach 13000 .

Step 1 ：We are given two data points (20, 900) and (35, 1400) which represent the population of the bacteria culture at different times. We can use these data points to solve for the initial population (P0) and the growth rate (k) in the exponential growth model P(t) = P0 * e^(kt).

Step 2 ：By substituting the given data points into the equation, we can form two equations and solve for P0 and k. After solving, we find that k = 0.029455516818602614 and P0 = 499.3386020983696.

Step 3 ：Using these values, we can find the doubling period (T_double) by using the formula T_double = ln(2)/k. Substituting the value of k, we find that T_double = 23.531998600757486 minutes.

Step 4 ：We can also find the population after 85 minutes (P_85) by substituting t = 85 into the equation P(t) = P0 * e^(kt). After substituting the values of P0 and k, we find that P_85 = 6105.854576587008.

Step 5 ：Finally, we can find the time when the population will reach 13000 (t_13000) by setting P(t) = 13000 and solving for t. After substituting the values of P0 and k, we find that t_13000 = 110.65567885174333 minutes.

Step 6 ：Final Answer: The initial size of the culture was approximately \(\boxed{499}\). The doubling period is approximately \(\boxed{23.53}\) minutes. The population after 85 minutes will be approximately \(\boxed{6106}\). The population will reach 13000 after approximately \(\boxed{110.66}\) minutes.