Let's break down the problem step-by-step to determine the correct statement related to the given population function [tex]\( C(B) = A(1 + r)^t \)[/tex].
1. Understanding the variables and the formula:
- [tex]\( C(B) \)[/tex]: This denotes the population at year [tex]\( t \)[/tex].
- [tex]\( A \)[/tex]: This is the initial population.
- [tex]\( r \)[/tex]: This is the rate of increase.
- [tex]\( t \)[/tex]: This is the time in years.
2. Analyzing each statement:
- Statement A: "The value of [tex]\( C \)[/tex] depends only on the starting population and the year."
- This statement overlooks the rate of increase [tex]\( r \)[/tex]. Clearly, the value of [tex]\( C(B) \)[/tex] depends on not just the initial population [tex]\( A \)[/tex] and time [tex]\( t \)[/tex], but also the rate [tex]\( r \)[/tex]. Therefore, this statement is not correct.
- Statement B: "P represents the population in year 0."
- In the given context, [tex]\( P \)[/tex] is not defined in the formula. Instead, the initial population is denoted by [tex]\( A \)[/tex]. Thus, this statement is incorrect.
- Statement C: "r is less than 0."
- The formula does not imply any specific restrictions on [tex]\( r \)[/tex]. While [tex]\( r \)[/tex] being negative would imply a declining population, this is not a general rule inferred from the function provided. Hence, this statement is not necessarily correct.
- Statement D: "The factor [tex]\( (1 + r) \)[/tex] is the percentage by which the population grows each year."
- This statement correctly interprets the growth factor. The factor [tex]\( (1 + r) \)[/tex] indeed represents the multiplicative growth rate each year, where [tex]\( r \)[/tex] is the annual rate of increase. So, if [tex]\( r \)[/tex] is the growth rate per year, [tex]\( (1 + r) \)[/tex] signifies the multiplication factor for the population each year. This statement is correct.
3. Conclusion:
- Upon evaluating each statement, the correct statement is D: "The factor [tex]\( (1 + r) \)[/tex] is the percentage by which the population grows each year."
Thus, the correct statement is:
[tex]\[ \boxed{\text{D}} \][/tex]
