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An experiment observing the growth of two germ strand populations finds these patterns:
- The population of the first germ is represented by the function [tex]A(x) = (1.3)^{x+9}[/tex].
- The population of the second germ is represented by the function [tex]B(x) = (1.3)^{4x+1}[/tex].

Which function best represents [tex]R(x)[/tex], the ratio of the population of the second germ to the population of the first germ?

A. [tex]R(x) = (2.6)^{3x-8}[/tex]

B. [tex]R(x) = (1.3)^{5x+10}[/tex]

C. [tex]R(x) = (2.6)^{4x^2+9}[/tex]

D. [tex]R(x) = (1.3)^{3x-8}[/tex]


Sagot :

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]