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Let's work through the problem step by step. The population of the Bulbul birds is modeled by the equation:

\[
P = 1350 + 400(1.25)^{-t}
\]

where \( P \) is the population of the birds, and \( t \) is the number of days since the start of the migration season. Let's address each part of the problem:

### (a) Find the population of the Bulbul birds at the start of the migration season.

At the start of the migration season, \( t = 0 \).

\[
P(0) = 1350 + 400(1.25)^{-0} = 1350 + 400(1) = 1350 + 400 = 1750
\]

So, the population of the birds at the start of the migration season is **1750**.

---

### (b) Find the population of the Bulbul birds after 5 days.

Here, we substitute \( t = 5 \) into the equation:

\[
P(5) = 1350 + 400(1.25)^{-5}
\]

We can calculate this value next.

---

### (c) On which day will the population decrease below 1400 for the first time?

We need to solve for \( t \) when \( P(t) = 1400 \):

\[
1400 = 1350 + 400(1.25)^{-t}
\]

We can rearrange this equation and solve for \( t \) algebraically.

---

### (d) Find the smallest possible population of Bulbul birds during the migration season.

The smallest population occurs as \( t \to \infty \), since \( (1.25)^{-t} \) will approach 0. Therefore, the smallest possible population is simply \( 1350 \), as the exponential term becomes negligible.

---

I'll calculate the remaining values now.### (b) The population of the Bulbul birds after 5 days is approximately **1481**.

### (c) The solution for \( t \) where the population falls below 1400 gives a negative result, which suggests that the population never decreases below 1400 in this model. The closest population approaches 1400 but doesn't reach or go below it.

Would you like more details on any of these steps? Here are 5 follow-up questions to expand understanding:

1. How is exponential decay reflected in the population model?
2. Why does the population never drop below 1400?
3. Can you explain the behavior of the population as \( t \to \infty \)?
4. How does changing the base of the exponent, 1.25, affect the rate of decay?
5. What is the significance of the constant 1350 in the population equation?

**Tip:** In exponential models, the base of the exponent controls the growth or decay rate. When the base is greater than 1, the quantity grows; when it is between 0 and 1, the quantity decays.

Professor Milioni investigated the migration season of the Bulbul bird from their natural wetlands to a warmer climate. She found that during the migration season their population, P, could be modelled by P = 1350 + 400(1.25)^-t, t ≥ 0. The tasks are to find the population at the start and after 5 days, determine the day the population first decreases below 1400, and find the smallest possible population during the season.

Explore how the Bulbul bird population changes during migration using exponential decay functions. This example calculates initial and future populations, identifies when the population falls below a specific threshold, and finds the minimum population possible according to the model.

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