the population of a village is 6400. the population is decreasing exponentially at a rate of x per year. after 22 years the population is 2607. find the value of x

This problem involves exponential decay. The population of the village decreases according to the equation:

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

Where:
- \( P(t) \) is the population after time \( t \),
- \( P_0 \) is the initial population,
- \( x \) is the rate of decay,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.

Given:
- Initial population \( P_0 = 6400 \),
- Population after 22 years \( P(22) = 2607 \),
- Time \( t = 22 \).

We substitute these values into the formula:

\[
2607 = 6400 e^{-22x}
\]

To solve for \( x \), we first divide both sides of the equation by 6400:

\[
\frac{2607}{6400} = e^{-22x}
\]

Now take the natural logarithm (ln) of both sides to eliminate the exponential function:

\[
\ln \left( \frac{2607}{6400} \right) = -22x
\]

Next, solve for \( x \):

\[
x = -\frac{\ln \left( \frac{2607}{6400} \right)}{22}
\]

Let me calculate this for you.The value of \( x \), the annual rate of population decrease, is approximately \( 0.0408 \) or 4.08% per year.

Would you like more details on how this solution was derived, or do you have any questions?

Here are 5 related questions to explore further:
1. How would the equation change if the population was increasing instead of decreasing?
2. What would the population be after 30 years at this rate?
3. How would you calculate the time needed for the population to halve?
4. Can you derive the rate of decay if given the population after a different number of years?
5. How does the natural logarithm help in solving exponential decay problems?

**Tip:** When solving exponential equations, using logarithms is crucial for eliminating the exponential terms.

The population of a village is 6400. The population is decreasing exponentially at a rate of x per year. After 22 years, the population is 2607. Find the value of x.

Learn how to calculate the exponential decay rate of a village's population, which decreases from 6400 to 2607 in 22 years. Using key concepts like exponential decay and logarithms, this problem is solved step-by-step. Suitable for Grades 10-12.

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