Question: Question 14
The population of a small town is decreasing at a rate of 3% each year. The
following table shows a projection of the population, $y$, after $t$ years.
Years Population
0 8,500
1 8,245
2 7,998
3 7,758
4 7,525
If the population of the small town is currently 8,500 people, how many years
will it take for the population to reach 5,500 people?
13.3 years
14.7 years
13.7 years
14.3 years
1. **State the problem:** We want to find how many years, $t$, it will take for the population to decrease from 8,500 to 5,500 given a 3% annual decrease.
2. **Formula used:** The population decreases exponentially, so we use the formula for exponential decay:
$$y = y_0 (1 - r)^t$$
where:
- $y$ is the population after $t$ years,
- $y_0 = 8500$ is the initial population,
- $r = 0.03$ is the rate of decrease (3%),
- $t$ is the number of years.
3. **Set up the equation:** We want $y = 5500$, so:
$$5500 = 8500 (1 - 0.03)^t = 8500 (0.97)^t$$
4. **Isolate $(0.97)^t$:**
$$\frac{5500}{8500} = (0.97)^t$$
$$\frac{\cancel{5500}}{\cancel{8500}} = (0.97)^t$$
Simplify the fraction:
$$\frac{5500}{8500} = \frac{55}{85} = \frac{11}{17} \approx 0.6471$$
So:
$$(0.97)^t = 0.6471$$
5. **Solve for $t$ using logarithms:**
Take natural logarithm on both sides:
$$\ln((0.97)^t) = \ln(0.6471)$$
Use the power rule of logarithms:
$$t \ln(0.97) = \ln(0.6471)$$
6. **Calculate $t$:**
$$t = \frac{\ln(0.6471)}{\ln(0.97)}$$
Calculate the values:
$$\ln(0.6471) \approx -0.4352$$
$$\ln(0.97) \approx -0.03046$$
So:
$$t = \frac{-0.4352}{-0.03046} \approx 14.29$$
7. **Interpretation:** It will take approximately 14.3 years for the population to reach 5,500.
**Final answer:** 14.3 years