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. The number of new 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 can be modeled using exponential growth. The general formula for exponential growth is:

\[
A(t) = A_0 \cdot (1 + r)^t
\]

Where:
- \( A(t) \) is the amount after time \( t \).
- \( A_0 \) is the initial amount (1000 people).
- \( r \) is the growth rate per time period (40% increase every 10 days, or 0.40 as a decimal).
- \( t \) is the number of time periods (each time period is 10 days).

Given that the campaign runs for 100 days, the number of periods \( t \) will be:

\[
t = \frac{100}{10} = 10
\]

Now, substituting the values into the formula:

\[
A(100) = 1000 \cdot (1 + 0.40)^{10}
\]

\[
A(100) = 1000 \cdot (1.40)^{10}
\]

Let’s calculate \( (1.40)^{10} \) and find the total number of people aware after 100 days.The total number of people who will be aware of the product after 100 days is approximately 28,925.

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What if the growth rate were different, say 30% instead of 40%?
2. How many people would be aware after 50 days instead of 100 days?
3. What is the formula for continuous growth, and how does it apply here?
4. How does exponential growth differ from linear growth in this context?
5. Can this model be used to predict awareness beyond 100 days, and what assumptions must hold?

**Tip:** Exponential growth can lead to surprisingly large numbers over time, especially with higher growth rates!

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. The number of new 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.

Discover how to calculate the total number of people aware of a product after 100 days of an advertising campaign using exponential growth. Starting with 1000 people and a 40% increase every 10 days, find out how this powerful mathematical concept leads to nearly 29,000 people being aware. Perfect for students in Grades 9-12.

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