For each year, \( t \), the population of a forest of trees, call it Forest \( A \), is represented by the function \( A(t)=111(1.025)^{t} \). In a neighboring forest, call it Forest \( B \), the population of the same type of tree is represented by the function \( B(t)=82(1.03)^{t} \). \( a \). Which forest's population is growing at a faster rate? Select an answer \( \hat{v} \) \( b \). Which forest had a greater number of trees initially? Select an answer \( \hat{v} \) By how many (round to the nearest tree)? trees \( c \). Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 50 years (round to the nearest tree)? Select an answer \( \hat{v} \) By how many?

Let's analyze each part of the problem step by step. ### **a. Which forest's population is growing at a faster rate?** - **Forest A:** \( A(t) = 111(1.025)^t \) - **Growth Rate:** 2.5% per year - **Forest B:** \( B(t) = 82(1.03)^t \) - **Growth Rate:** 3.0% per year **Answer:** **Forest B** is growing at a faster rate because its annual growth rate of **3.0%** is higher than Forest A's **2.5%**. --- ### **b. Which forest had a greater number of trees initially? By how many?** - **Initial Population of Forest A:** \( A(0) = 111 \) trees - **Initial Population of Forest B:** \( B(0) = 82 \) trees **Answer:** **Forest A** had a greater number of trees initially. **Difference:** \( 111 - 82 = 29 \) trees --- ### **c. Which forest will have a greater number of trees after 50 years? By how many?** - **Population after 50 years:** - **Forest A:** \[ A(50) = 111 \times (1.025)^{50} \approx 111 \times 3.434 \approx 381 \text{ trees} \] - **Forest B:** \[ B(50) = 82 \times (1.03)^{50} \approx 82 \times 4.392 \approx 360 \text{ trees} \] **Answer:** After **50 years**, **Forest A** will have a greater number of trees. **Difference:** \( 381 - 360 = 21 \) trees --- ### **Summary of Answers:** a. **Forest B** is growing at a faster rate. b. **Forest A** had a greater number of trees initially by **29 trees**. c. After 50 years, **Forest A** will have more trees by **21 trees**.