Question

In a specific lake, the fish population is declining at a rate of 3.5% per year. If there are currently 1,200 fish in the lake, create an exponential decay function to model the problem. If the decrease in population continues at this rate, how many fish will be in the lake after 20 years? Round your answer to the nearest whole number.

Answer 1

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:

\[
\boxed{596}
\]