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To find the correct function \( P(t) \) that describes the world population growth, we need to interpret the information provided:

- The population growth rate is 1.3% per year since the year 2000.
- The initial population in the year 2000 was 6 billion.
- We are asked to find a function \( P(t) \), where \( t \) is the number of years after 2000.

A 1.3% growth rate is exponential, meaning the population grows by a factor each year, rather than adding a fixed amount. This suggests that an exponential model is most appropriate.

### Step-by-Step Solution
The general form for an exponential growth model is:
\[
P(t) = P_0 \cdot (1 + r)^t
\]
where:
- \( P_0 = 6 \) billion (the initial population),
- \( r = 0.013 \) (the growth rate of 1.3% per year),
- \( t \) is the number of years since 2000.

So, the correct function should be:
\[
P(t) = 6 \cdot (1.013)^t
\]

### Verification with Options
Looking at the options:
1. \( P(t) = 6 \cdot 1.3^t \): **Incorrect**. This represents a 30% growth rate, not 1.3%.
2. \( P(t) = 1.13t + 6 \): **Incorrect**. This represents a linear model, not an exponential model.
3. \( P(t) = 1.013t + 6 \): **Incorrect**. This is also a linear model.
4. \( P(t) = 6 \cdot 1.013^t \): **Correct**. This matches the exponential growth model with 1.3% growth.
5. \( P(t) = 0.013t + 4 \): **Incorrect**. This is a linear model and does not start with the correct initial population.

### Conclusion
The correct answer is:
\[
\boxed{P(t) = 6 \cdot 1.013^t}
\]

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How would the formula change if the growth rate was 2% instead of 1.3%?
2. What is the population after 10 years according to this model?
3. How does exponential growth differ from linear growth?
4. How would the function change if the starting year was 2010 with a different initial population?
5. Can this model be used indefinitely, or are there limitations to exponential growth?

**Tip:** Exponential models are ideal for situations where a quantity grows by a fixed percentage per time period.

Verdens befolkning er vokset med en gennemsnitlig rate på 1,3 procent pr. år siden år 2000. Der var 6 milliarder mennesker i verden i år 2000. Hvilken af følgende funktioner, P, beskriver verdens befolkning, i milliarder personer, t år efter år 2000?

Explore how to model world population growth with an exponential function given a 1.3% annual growth rate since the year 2000. Learn how to determine the correct exponential function to describe the population change over time. Suitable for students in Grades 10-12.

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