To model the decline of the fish population in the lake using an exponential decay function, we can use the formula:
\[ P(t) = P_0 \cdot e^{kt} \]
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,
- \(e\) is the base of the natural logarithm.
Given:
- \(P_0 = 1200\) (the initial number of fish),
- The fish population is declining at a rate of 3.5% per year, which means \(k = -0.035\) (the negative sign indicates a decrease).
The model can therefore be written as:
\[ P(t) = 1200 \cdot e^{-0.035t} \]
Now, we want to find the fish population after 20 years, so we will substitute \(t = 20\) into the equation.
\[ P(20) = 1200 \cdot e^{-0.035 \times 20} \]
Calculating the exponent:
\[ -0.035 \times 20 = -0.7 \]
Now substituting back into the equation:
\[ P(20) = 1200 \cdot e^{-0.7} \]
Next, we calculate \(e^{-0.7}\):
\[ e^{-0.7} \approx 0.496585 \]
Now multiply this by 1200:
\[ P(20) \approx 1200 \cdot 0.496585 \approx 595.902 \]
Rounding to the nearest whole number, we find:
\[ P(20) \approx 596 \]
Therefore, after 20 years, the fish population in the lake is expected to be approximately 596 fish.