If the population of a city is 145 comma 000.  If the population declines continuously by 4​% per​ year, what will the population be in 15 ​years?

The population of the city is declining continuously at a rate of 4% per year. This is a typical case of exponential decay, which can be modeled using the formula for continuous growth or decay:

\[
P(t) = P_0 e^{rt}
\]

Where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population,
- \(r\) is the rate of decline (negative for decay),
- \(t\) is the time in years,
- \(e\) is Euler's number (approximately 2.71828).

Given:
- \(P_0 = 145,000\),
- \(r = -0.04\) (since the population is declining by 4% per year),
- \(t = 15\) years.

Now, we can plug these values into the formula to calculate the population after 15 years:

\[
P(15) = 145,000 \cdot e^{-0.04 \cdot 15}
\]

Let me calculate that.The population of the city after 15 years will be approximately 79,578.

Would you like more details or have any questions?

Here are 5 related questions:
1. What would the population be after 20 years at the same decline rate?
2. How does the formula change if the population increases instead of declining?
3. How would a different decline rate, such as 6% per year, affect the population after 15 years?
4. How is continuous decay different from discrete decay?
5. What would happen if the population declined by 4% every six months instead of yearly?

**Tip:** Continuous decay models are useful in many real-world scenarios, such as population decline, radioactive decay, and depreciation of assets.

If the population of a city is 145,000 and it declines continuously by 4% per year, what will the population be in 15 years?

Learn how to calculate the population decline of a city from 145,000 over 15 years using the exponential decay formula. This problem demonstrates continuous decay at a 4% yearly rate, essential for students studying exponential functions and population dynamics in Grades 10-12.

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