The number of fish in a lake is decreasing at the rate of 3% per year. In 2015, there were 2000 fish in the lake. Write an exponential decay function to model the number of fish over time. Use function to find the number of fish in the lake in 2025.

Answer 1

To model the number of fish in the lake using an exponential decay function, we can start with the general formula for exponential decay:

$$N(t)=N_0\times e^{-kt}$$

where:

• $N(t)$ is the quantity at time $t$,
• $N_0$ is the initial quantity,
• $k$ is the decay constant,
• $t$ is the time in years,
• $e$ is the base of the natural logarithm.

In this case, we have:

• $N_0=2000$ (the initial number of fish),
• the decay rate is 3%, which can be represented as a decay constant of $k=0.03$.

Thus, the exponential decay function for the number of fish in the lake can be modeled as:

$$N(t)=2000\times e^{-0.03t}$$

Next, we need to find the number of fish in the lake in 2025. Since 2015 is our starting point, we compute $t$ as follows:

$$t=2025-2015=10\text{ years}$$

Now we substitute $t=10$ into the decay function:

$$N(10)=2000\times e^{-0.03\times 10}$$

Calculating $-0.03\times 10$:

$$-0.03\times 10=-0.3$$

So we have:

$$N(10)=2000\times e^{-0.3}$$

Now we need to compute $e^{-0.3}$. Using a calculator, we find:

$$e^{-0.3}\approx 0.740818$$

Now we can plug this value back into the function:

$$N(10)\approx 2000\times 0.740818\approx 1481.636$$

Rounding to the nearest whole fish, we find:

$$N(10)\approx 1482$$

Thus, the estimated number of fish in the lake in 2025 is approximately 1482 fish.