An advertising company designs a campaign to introduce a new product to a metropolitan area. The company determines that 1000 people are aware of the product at the beginning of the campaign, and that the number of people aware increases by 40% every 10 days during the advertising campaign. Determine the total number of people who will be aware of the product after 100 days.

This problem involves exponential growth, where the number of people aware of the product increases by 40% every 10 days.

We can model this situation using the formula for exponential growth:

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

Where:
- \(P(t)\) is the population after time \(t\),
- \(P_0\) is the initial population (1000 people in this case),
- \(r\) is the growth rate (40% or 0.40),
- \(n\) is the number of growth periods.

### Step-by-step solution:

1. **Initial population**: \(P_0 = 1000\)
2. **Growth rate**: \(r = 0.40\)
3. **Time**: \(t = 100\) days.
4. **Each period lasts 10 days**, so the number of periods \(n = \frac{t}{10} = \frac{100}{10} = 10\).

Now we can apply the formula:

\[
P(100) = 1000 \times (1 + 0.40)^{10}
\]
\[
P(100) = 1000 \times (1.40)^{10}
\]

Next, calculate \( (1.40)^{10} \). Let me compute that.

After 100 days, approximately 28,925 people will be aware of the product.

Do you want further clarification on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. How would the result change if the growth rate was 50% every 10 days?
2. What would happen if the initial population was 2000 instead of 1000?
3. How can you calculate the population after 50 days instead of 100?
4. What would the equation look like if the growth happened every 5 days instead of 10?
5. How does this growth model compare to linear growth?

**Tip**: Exponential growth models are commonly used to represent processes where the rate of change is proportional to the current amount, such as in population growth and interest calculations.

Learn how to calculate the total number of people aware of a product after 100 days, starting from 1000 people, with a 40% increase every 10 days. Explore exponential growth concepts with real-world applications. Suitable for Grades 10-12 students.

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