The size \( P \) of a certain insect population at time \( t \) (in days) obeys the function \( P(t)=600 e^{0.04 t} \) (a) Determine the number of insects at \( t=0 \) days. (b) What is the growth rate of the insect population? (c) Graph the function using a graphing utility (d) What is the population after 10 days? (e) When will the insect population reach 720 ? (f) When will the insect population double?

To help you with a bit of extra context, let's dive in! At \( t=0 \) days, we can find the initial population by simply plugging in \( t=0 \) into the function: \( P(0) = 600 e^{0.04 \cdot 0} = 600 e^0 = 600 \). So, there are 600 insects at the start. As for the growth rate, the term \( 0.04 \) in the exponent indicates that the population grows continuously at a rate of 4% per day. Now, isn't it fascinating how much insects can proliferate under optimal conditions? They can multiply quickly, which is why understanding their growth function is crucial for managing ecosystems or even agricultural pests! If you're hungry for more detailed insights, consider reading about exponential growth models in biology. These models aren't exclusive to insects; they're relevant in fields like finance and technology, illustrating how systems can escalate rapidly when left unchecked! Exploring real-world applications shows just how interconnected these concepts are, leading to unique insights in diverse fields.