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The problem in the image asks:

**"If the world population increases by 2% per year, how many years will it take for the population to double?"**

This is a classic case of **exponential growth**. We can solve this using the formula for exponential growth:

\[
P(t) = P_0 \times (1 + r)^t
\]

Where:
- \( P(t) \) is the population after time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the annual growth rate (in decimal form),
- \( t \) is the number of years.

We are looking for the time \( t \) when the population doubles, meaning \( P(t) = 2 \times P_0 \). The growth rate \( r \) is 2%, or \( 0.02 \).

We can set up the equation as:

\[
2P_0 = P_0 \times (1 + 0.02)^t
\]

Dividing both sides by \( P_0 \):

\[
2 = (1.02)^t
\]

Now, we solve for \( t \) by taking the natural logarithm (ln) on both sides:

\[
\ln(2) = t \times \ln(1.02)
\]

Solving for \( t \):

\[
t = \frac{\ln(2)}{\ln(1.02)}
\]

Let me calculate this value.The time it will take for the population to double, given an annual growth rate of 2%, is approximately **35 years**.

Would you like any further details or clarification?

Here are some related questions you might find useful:
1. How would the time change if the growth rate were 3% instead of 2%?
2. What if the population needed to triple instead of double—how would that affect the time?
3. How does exponential growth compare to linear growth in terms of population models?
4. How can this formula be applied to other fields, such as finance or biology?
5. What would happen if the growth rate varied each year?

**Tip:** Understanding exponential growth is crucial for analyzing trends in population, finance, and even technology.

Si la population mondiale augmente de 2% par an, combien faudra-t-il d'années pour que la population double?

Learn how to calculate the time it takes for the world population to double with an annual growth rate of 2%. This problem uses key exponential growth formulas and logarithmic properties. Suitable for Grades 10-12.

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