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The problem in the image involves exponential population growth, where the population of Loreto follows the equation:

\[
P(t) = A_0 e^{kT}
\]

Where:
- \( A_0 \) is the initial population (in 1940, \( A_0 = 100,000 \)),
- \( T \) is the number of years after 1940,
- \( k \) is the growth constant (to be determined),
- \( P(t) \) is the population at time \( t \).

### Step 1: Determine \( k \)
Given:
- In 1940 (\( T = 0 \)), the population was 100,000.
- In 1970 (\( T = 30 \)), the population was 140,000.

Using the population equation at \( T = 30 \):
\[
140,000 = 100,000 e^{30k}
\]

Solving for \( k \):
\[
\frac{140,000}{100,000} = e^{30k}
\]
\[
1.4 = e^{30k}
\]
Taking the natural logarithm of both sides:
\[
\ln(1.4) = 30k
\]
\[
k = \frac{\ln(1.4)}{30}
\]

### Step 2: Calculate \( P(t) \) for the year 2000
In 2000, \( T = 60 \) (since 2000 is 60 years after 1940). We can now substitute the value of \( k \) and \( T = 60 \) into the exponential growth equation to find the population in 2000.

I'll calculate these values for you.The population of Loreto in the year 2000, according to the exponential growth model, was approximately 196,000 inhabitants.

Would you like any further details on this calculation or have any additional questions?

Here are 5 related questions that might interest you:
1. How would the population projection change if the growth rate was higher?
2. What would be the population in 2025 using the same growth model?
3. How can we calculate the doubling time of a population based on the growth rate?
4. What happens to the model if the population starts declining? How would the formula change?
5. How can we apply this model to predict populations in smaller intervals, such as every 5 years?

**Tip**: Exponential growth can quickly lead to large numbers, which is why it's often important to consider limiting factors in real-world applications, such as resources or space.

En 1940 la población de la ciudad de Loreto era de 100 000 habitantes. En 1970 había 140 000 habitantes. Asumamos que el número de habitantes en función del tiempo se ajusta a la fórmula: P(t) = A0 * e^(kT), donde A0 y k son constantes. ¿Cuál fue la población en 2000?

This problem explores exponential growth in population modeling. Learn how to calculate the population of Loreto in 2000 using the exponential growth equation P(t) = A0 * e^(kT). The problem provides step-by-step calculations for determining the growth rate constant (k) and projecting population values.

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