To determine which function best represents [tex]\( R(x) \)[/tex], the ratio of the population of the second germ [tex]\( B(x) \)[/tex] to the population of the first germ [tex]\( A(x) \)[/tex], we follow these steps:
1. Write down the given functions:
[tex]\[
A(x) = (1.3)^{x+9}
\][/tex]
[tex]\[
B(x) = (1.3)^{4x+1}
\][/tex]
2. Define the ratio [tex]\( R(x) \)[/tex]:
[tex]\[
R(x) = \frac{B(x)}{A(x)}
\][/tex]
3. Substitute [tex]\( B(x) \)[/tex] and [tex]\( A(x) \)[/tex] into the ratio:
[tex]\[
R(x) = \frac{(1.3)^{4x+1}}{(1.3)^{x+9}}
\][/tex]
4. Simplify the expression using properties of exponents:
[tex]\[
R(x) = \frac{(1.3)^{4x+1}}{(1.3)^{x+9}} = (1.3)^{(4x+1) - (x+9)}
\][/tex]
5. Combine the exponents:
[tex]\[
R(x) = (1.3)^{(4x + 1 - x - 9)} = (1.3)^{(4x - x + 1 - 9)} = (1.3)^{3x - 8}
\][/tex]
6. Identify the correct option:
[tex]\[
\boxed{R(x) = (1.3)^{3x-8}}
\][/tex]
Thus, the correct answer is:
D. [tex]\( R(x) = (1.3)^{3x-8} \)[/tex]
