Questions: 12. A population of fish starts at 8,000 and decreases by 6% per year. What is the population after 10 years? a) 14,327 b) 4309 c) 839 d) 7680

Transcript text: 12. A population of fish starts at 8,000 and decreases by 6\% per year. What is the population after 10 years? a) 14,327 b) 4309 c) 839 d) 7680

Solution

Solution Steps

Step 1: Define Initial Values

Let \( P_0 = 8000 \) be the initial population of fish, \( r = 0.06 \) be the rate of decrease per year, and \( t = 10 \) be the number of years.

Step 2: Apply the Exponential Decay Formula

Use the formula for exponential decay: \[ P(t) = P_0 \times (1 - r)^t \] Substituting the known values, we have: \[ P(10) = 8000 \times (1 - 0.06)^{10} \]

Step 3: Calculate the Population After 10 Years

Calculate \( (1 - 0.06)^{10} \): \[ P(10) = 8000 \times (0.94)^{10} \] Now compute the final population: \[ P(10) \approx 8000 \times 0.4309 \approx 4308.92 \]

Thus, the population of fish after 10 years is approximately \( 4308.92 \).