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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]
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]
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