Final answer:
Using the provided mathematical model, the projected population size of the insect species after 1 year (365 days) is 11834. The model suggests the population size could increase indefinitely, but this is usually limited by factors such as food, space etc. that form the environment's carrying capacity.
Step-by-step explanation:
The student's question is related to the field of Mathematics, specifically to exponential growth functions. The given function is P(t) = 65(1 + 0.5t) / (2 + 0.01t). To find out the projected size of the colony after 1 year (365 days), we need to replace 't' in the function with 365. Similarly, to find the maximum population that the protected area can sustain, we need to take the limit of the function as t approaches infinity.
By plugging '365' into the equation, and rounding to the whole number, we get: P(365) = 11834.
For finding a limit as t approaches infinity, recognizing that 65*(1+0.5t) grows much faster than (2+0.01t), indicating that, as t gets larger, the impact of '2' and '0.01t' in the denominator becomes negligible compared to the numerator. Thus, the function trends towards infinity, meaning the population size will indefinitely increase, given the model. For such scenarios, generally in ecology or environment science, the concept of carrying capacity is introduced which is the maximum population size that the environment can sustain indefinitely.
Learn more about Exponential Growth