To solve this problem, we’ll use the exponential decay formula:

$P(t) = P_0 \cdot (1 - r)^t$

Where:

• $P_0$ = initial population = 48,000
• $r$ = decay rate = 3% = 0.03
• $t$ = time in years = 7 years
• $P(t)$ = population after $t$ years

Step 1: Write the Exponential Decay Function

Substitute the known values into the formula:

$P(t) = 48,000 \cdot (1 - 0.03)^t$

Simplify $(1 - 0.03)$:

$P(t) = 48,000 \cdot (0.97)^t$

Step 2: Find the Population After 7 Years

Substitute $t = 7$ into the formula:

$P(7) = 48,000 \cdot (0.97)^7$

First, calculate $(0.97)^7$:

$(0.97)^7 \approx 0.868$

Now multiply:

$P(7) = 48,000 \cdot 0.868$

$P(7) \approx 41,664$

Final Answer:

The fish population after 7 years is approximately 41,664.

Let me know if you'd like further clarification! Here are 5 related questions you might explore:

1. How would the population change if the decay rate were 5% instead of 3%?
2. What is the population after 10 years under the same conditions?
3. Can you derive a general exponential decay formula for any $P_0$ and $r$?
4. How long will it take for the population to decrease to half its original size?
5. What would happen to the population if there were no decay (0% rate)?

Tip: In exponential decay, the population decreases rapidly at first, then slows down over time, following a curve. Understanding this helps in modeling real-world phenomena like radioactive decay and population loss.