Question

The population of a town is decreasing at a rate of 2% per year. In 2000 there were 1100 people. Write an exponential decay function to model this situation. Then find the population in 2009.

Answer 1

To write an exponential decay function to model the population of the town, we can start with the general formula for exponential decay:

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

Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant,
- \( t \) is the time in years.

In this case:

- The initial population \( P_0 = 1100 \).
- The population is decreasing at a rate of 2% per year, which means \( k = -0.02 \) (the negative sign indicates decay).

Now we can write the function:

\[ P(t) = 1100 e^{-0.02t} \]

Next, we want to find the population in the year 2009. Since we start counting our time \( t \) from the year 2000, the year 2009 corresponds to \( t = 9 \).

Now we substitute \( t = 9 \) into the function:

\[ P(9) = 1100 e^{-0.02 \times 9} \]

Calculating the exponent:

\[ -0.02 \times 9 = -0.18 \]

Now we find \( e^{-0.18} \). Using a calculator:

\[ e^{-0.18} \approx 0.8353 \]

Putting it all together:

\[ P(9) = 1100 \times 0.8353 \]

Calculating that gives:

\[ P(9) \approx 919.83 \]

Since the population must be a whole number, we round it to the nearest person:

\[ P(9) \approx 920 \]

Thus, the population in the year 2009 is approximately **920 people**.