Consider this scenario: For each year $t$, the population of a forest of trees is represented by the function $A(t)=114(1.021)^{t}$. In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=88(1.029)^{t}$.
Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 20 years?
forest represented by $A(t)$
forest represented by $B(t)$
By how many? (Round your answer to the nearest whole number.)
Answer
So, the forest represented by \(B(t)\) will have a greater number of trees after 20 years. The difference in the number of trees will be approximately \(\boxed{17}\).
Step 1 ：Let's calculate the population of trees in both forests after 20 years. For the first forest, we substitute \(t=20\) in the function \(A(t)=114(1.021)^{t}\). For the second forest, we substitute \(t=20\) in the function \(B(t)=88(1.029)^{t}\).
Step 2 ：After calculating, we find that the population of the first forest after 20 years is approximately 172.75, and the population of the second forest is approximately 155.88.
Step 3 ：To find out which forest will have a greater number of trees after 20 years, we compare the two populations. We find that the first forest will have more trees.
Step 4 ：To find out by how many, we subtract the smaller population from the larger one. The difference is approximately 17.
Step 5 ：So, the forest represented by \(B(t)\) will have a greater number of trees after 20 years. The difference in the number of trees will be approximately \(\boxed{17}\).
