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The graph provided represents the population \( p \) of fruit flies over time (in days), with the horizontal axis labeled as "Tiempo (días)" and the vertical axis representing the population \( p \).

Let’s break down and analyze the two questions:

### 1. **Determine la población que habrá de este insecto dentro de 20 y 45 días.**

To answer this, observe the graph at \( t = 20 \) and \( t = 45 \):

- At \( t = 20 \), the population appears to be approximately 50.
- At \( t = 45 \), the population is around 350 (the graph plateaus near this value after day 45).

Thus, between days 20 and 45, the population grows from about 50 to 350.

### 2. **¿Cuál es el rango (aproximado) de la función de población en el intervalo \( 0 \leq t \leq 50 \)?**

In the interval from \( t = 0 \) to \( t = 50 \), the graph shows:
- At \( t = 0 \), the population starts close to 0.
- By \( t = 50 \), the population reaches a maximum value of around 350.

Thus, the approximate range of the population function in this interval is from \( p = 0 \) to \( p = 350 \).

Would you like further details or clarifications on any part? Here are some related questions to explore:

1. How would the growth rate of the population be described mathematically between days 20 and 45?
2. What kind of function might model the population growth shown in the graph?
3. How does the population behave before day 20 and after day 45, based on the graph?
4. Why might the population stabilize after reaching 350?
5. How can we estimate the average population growth rate between days 0 and 50?

**Tip:** To model population growth, logistic functions are often used when there is an initial period of rapid growth followed by stabilization.

Determine la población que habrá de este insecto dentro de 20 y 45 días. ¿Cuál es el rango aproximado de la función de población en el intervalo 0 ≤ t ≤ 50?

Analyze the population growth of fruit flies over 50 days with a detailed interpretation of the graph. Understand key concepts like population range and growth behavior using mathematical models. Suitable for high school students.

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