To create an exponential decay function to model the fish population, we can use the formula:
\[ P(t) = P_0 e^{kt} \]
Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant,
- \( t \) is time in years,
- \( e \) is the base of the natural logarithm.
In this case, the initial fish population \( P_0 \) is 1,200 fish. The population is declining at a rate of 3.5% per year, which means \( k \) will be negative. Specifically, since the decay rate is 3.5%, we have:
\[ k = -0.035 \]
Substituting these values into the exponential decay model, we get:
\[ P(t) = 1200 e^{-0.035t} \]
Next, we want to find the fish population after 20 years (\( t = 20 \)).
Substituting \( t = 20 \) into the equation:
\[ P(20) = 1200 e^{-0.035 \times 20} \]
Calculating the exponent:
\[ -0.035 \times 20 = -0.7 \]
Thus, we have:
\[ P(20) = 1200 e^{-0.7} \]
Now, we calculate \( e^{-0.7} \):
Using a calculator, we find:
\[ e^{-0.7} \approx 0.496585 \]
Now, substituting this value back into the function:
\[ P(20) = 1200 \times 0.496585 \] \[ P(20) \approx 595.902 \]
Rounding to the nearest whole number, we find that the estimated fish population after 20 years is:
\[ \boxed{596} \]
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