To solve this problem, we will first set up the exponential decay function and then calculate the population after 7 years.

1. Exponential Decay Formula

The exponential decay model is given by:

$P(t) = P_0 \cdot e^{-rt}$

Where:

• $P(t)$ = population after $t$ years
• $P_0$ = initial population
• $r$ = decay rate (as a decimal)
• $t$ = time in years

2. Known Values

• $P_0 = 48000$ (initial population)
• $r = 3\% = 0.03$ (decay rate as a decimal)
• $t = 7$ years

3. Substituting the Values

Substitute into the decay model:

$P(7) = 48000 \cdot e^{-0.03 \cdot 7}$

Simplify the exponent:

$P(7) = 48000 \cdot e^{-0.21}$

Using a calculator or the exponential function:

$e^{-0.21} \approx 0.8106$

Now multiply:

$P(7) = 48000 \cdot 0.8106 \approx 38908.8$

4. Final Answer

The population after 7 years is approximately 38,909 fish.

Do you want further details or have any questions?

Here are 5 related questions to extend your understanding:

1. How does the exponential decay model change if the decay rate increases?
2. What would the population be after 10 years instead of 7?
3. Can we determine the time it takes for the population to decrease by half (half-life)?
4. How is the formula modified if the decay is not continuous but yearly?
5. How would the model look if the population were increasing instead of decreasing?

Tip: When solving exponential growth or decay problems, always convert percentages into decimals before substitution.