There are approximately 3,000 bass in a lake. The population grows at a rate of \( 2 \% \) per year. From Year 1 to Year 4 , the average rate of change of the population to the nearest tenth was about From Year 5 to Year 8 , the average rate of change of the population to the nearest tenth was about The rate of change increased by about bass per a year.

Let's analyze the population growth of bass in the lake step by step. ### Given: - **Initial Population (P₀):** 3,000 bass - **Growth Rate:** 2% per year - **Population Function:** \( P(t) = 3000 \times (1.02)^t \), where \( t \) is the number of years ### 1. Average Rate of Change from Year 1 to Year 4 **Calculate Population at Year 1 and Year 4:** - **Year 1 (P₁):** \[ P(1) = 3000 \times 1.02^1 = 3000 \times 1.02 = 3060 \text{ bass} \] - **Year 4 (P₄):** \[ P(4) = 3000 \times 1.02^4 \approx 3000 \times 1.08243216 \approx 3247.3 \text{ bass} \] **Average Rate of Change:** \[ \text{Average Rate} = \frac{P(4) - P(1)}{4 - 1} = \frac{3247.3 - 3060}{3} \approx \frac{187.3}{3} \approx 62.4 \text{ bass per year} \] ### 2. Average Rate of Change from Year 5 to Year 8 **Calculate Population at Year 5 and Year 8:** - **Year 5 (P₅):** \[ P(5) = 3000 \times 1.02^5 \approx 3000 \times 1.1040808 \approx 3312.2 \text{ bass} \] - **Year 8 (P₈):** \[ P(8) = 3000 \times 1.02^8 \approx 3000 \times 1.171659 \approx 3515.0 \text{ bass} \] **Average Rate of Change:** \[ \text{Average Rate} = \frac{P(8) - P(5)}{8 - 5} = \frac{3515.0 - 3312.2}{3} \approx \frac{202.8}{3} \approx 67.6 \text{ bass per year} \] ### 3. Increase in the Rate of Change \[ \text{Increase} = 67.6 - 62.4 = 5.2 \text{ bass per year} \] ### **Summary:** - **From Year 1 to Year 4:** The average rate of change was **62.4 bass per year**. - **From Year 5 to Year 8:** The average rate of change was **67.6 bass per year**. - **Increase in Rate of Change:** **5.2 bass per year**.