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The question is about modeling an insect population that grows exponentially, with an initial population and a known tripling period. 

### Part (a): 
We are asked to write an exponential function of the form \( P(t) = P_0 \left(3^{\frac{t}{n}}\right) \) that models the population, where:
- \( P_0 \) is the initial population,
- \( t \) is the time in days,
- \( n \) is the time it takes for the population to triple.

#### Given:
- The population triples in 30 days (so \( n = 30 \)),
- The initial population \( P_0 = 40 \) insects.

The general form for the population function is:

\[
P(t) = P_0 \left( 3^{\frac{t}{n}} \right)
\]

Now, substituting the given values:

\[
P(t) = 40 \left( 3^{\frac{t}{30}} \right)
\]

This equation models the population at any time \( t \).

### Final answer:
\[
P(t) = 40 \cdot 3^{\frac{t}{30}}
\]

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Would you like further details on the exponential model, or have any specific questions?

### Related questions:
1. What is the population after 15 days using the model \( P(t) = 40 \cdot 3^{\frac{t}{30}} \)?
2. How long will it take for the population to double instead of triple?
3. What would the equation be if the population were to quadruple every 30 days?
4. How would the function change if the initial population were different?
5. What is the population growth rate implied by this model?

### Tip:
Exponential growth functions are powerful for modeling populations, as they grow by constant factors over equal time intervals.

If the population triples in 30 days, and 40 insects are present initially, write an exponential function of the form P(t) = P0(3^(t/n)) that models the population.

Learn how to model an exponentially growing insect population that triples in 30 days, starting with 40 insects. This problem explores exponential functions in real-world applications and is suitable for Grades 10-12.

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