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 your question step-by-step. --- ### **a. Which forest's population is growing at a faster rate?** To determine which forest has a faster growth rate, we compare the growth factors in their respective population functions: - **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% per year **Conclusion:** **Forest B** has a faster population growth rate at **3% per year** compared to Forest A's **2.5% per year**. --- ### **b. Which forest had a greater number of trees initially? By how many?** To find out which forest had more trees initially, evaluate each function at \( t = 0 \): - **Forest A:** \[ A(0) = 111(1.025)^0 = 111 \times 1 = 111 \text{ trees} \] - **Forest B:** \[ B(0) = 82(1.03)^0 = 82 \times 1 = 82 \text{ trees} \] **Difference:** \[ 111 \text{ (Forest A)} - 82 \text{ (Forest B)} = 29 \text{ trees} \] **Conclusion:** **Forest A** initially had more trees, with **29 more trees** than Forest B. --- ### **c. Which forest will have a greater number of trees after 50 years? By how many?** We'll calculate the populations of both forests after 50 years using their respective growth functions. 1. **Calculate \( A(50) \):** \[ A(50) = 111 \times (1.025)^{50} \] - Approximating \( (1.025)^{50} \approx 3.4368 \) - \( A(50) \approx 111 \times 3.4368 \approx 381 \) trees 2. **Calculate \( B(50) \):** \[ B(50) = 82 \times (1.03)^{50} \] - Approximating \( (1.03)^{50} \approx 4.381 \) - \( B(50) \approx 82 \times 4.381 \approx 359 \) trees **Difference:** \[ 381 \text{ (Forest A)} - 359 \text{ (Forest B)} = 22 \text{ trees} \] **Conclusion:** After **50 years**, **Forest A** will have more trees than Forest B by approximately **22 trees**. --- ### **Summary of Answers:** - **a.** **Forest B** is growing at a faster rate. - **b.** **Forest A** initially had more trees by **29 trees**. - **c.** After 50 years, **Forest A** will have more trees by approximately **22 trees**.