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.
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