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?
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To determine which forest's population is growing at a faster rate, we can examine the growth rates represented by the coefficients of \( t \) in the functions. For Forest \( A \), the growth rate is \( 2.5\% \) (or \( 0.025 \)), while for Forest \( B \), it is \( 3\% \) (or \( 0.03 \)). Thus, the answer to part \( a \) is that Forest \( B \) is growing at a faster rate. For the initial populations, we see that Forest \( A \) has 111 trees and Forest \( B \) has 82 trees. Therefore, the answer to part \( b \) is that Forest \( A \) had a greater number of trees initially. Specifically, the difference is \( 111 - 82 = 29 \) trees. Now, let's project the populations after 50 years. For Forest \( A \): \[ A(50) = 111(1.025)^{50} \approx 111 \times 3.208 \approx 356 \text{ trees} \] For Forest \( B \): \[ B(50) = 82(1.03)^{50} \approx 82 \times 4.384 \approx 360 \text{ trees} \] Thus, for part \( c \), Forest \( B \) will have a greater number of trees after 50 years by approximately \( 360 - 356 = 4 \) trees. So the complete answers are: a. Forest \( B \), b. Forest \( A \), by 29 trees, c. Forest \( B \), by 4 trees.
