A town has a population of 76,000 and shrinks at a rate of 7% every year. Which equation represents the town’s population after 4 years?

The situation you're describing can be modeled using the exponential decay formula:

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

where:
- \( P(t) \) is the population after \( t \) years,
- \( P_0 \) is the initial population (76,000),
- \( r \) is the decay rate (7% or 0.07),
- \( t \) is the number of years (4 years in this case).

Substituting the values:

\[
P(4) = 76,000 \times (1 - 0.07)^4
\]

Simplifying:

\[
P(4) = 76,000 \times (0.93)^4
\]

This is the equation that represents the population after 4 years. You can further calculate the exact value if needed.

Would you like me to compute the final population?

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Here are 5 related questions:
1. What would the population be after 10 years at the same rate?
2. How does exponential decay differ from linear decay?
3. How would the equation change if the population increased by 7% each year?
4. Can you rewrite the formula for any arbitrary number of years \( t \)?
5. What factors in real life could affect the accuracy of such a model?

**Tip:** Exponential decay and growth processes occur frequently in finance, biology, and physics, so mastering the formula is very useful!

Learn how to model population decay using exponential decay. This example shows how a population of 76,000 shrinks by 7% each year, represented by the equation P(t) = P_0 (1 - r)^t after 4 years. Ideal for students in Grades 9-12.

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