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?

To determine which forest's population is growing at a faster rate, we compare the growth rates, or the coefficients of the exponential functions for each forest. In Forest \( A \), the growth rate is \( 1.025 \), while in Forest \( B \), it is \( 1.03 \). Since \( 1.03 > 1.025 \), Forest \( B \) is growing at a faster rate. For the initial populations, Forest \( A \) starts with \( 111 \) trees and Forest \( B \) starts with \( 82 \) trees. Therefore, Forest \( A \) had a greater number of trees initially by \( 111 - 82 = 29 \) trees. To find out which forest will have a greater number of trees after 50 years, we can calculate \( A(50) \) and \( B(50) \): - \( A(50) = 111(1.025)^{50} \approx 111(3.207) \approx 356 \). - \( B(50) = 82(1.03)^{50} \approx 82(4.384) \approx 360 \). Thus, after 50 years, Forest \( B \) will have more trees, approximately \( 360 - 356 = 4 \) trees more than Forest \( A \).