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Sagot :
To find [tex]\( F(2) \)[/tex] using the function [tex]\( F(t) = 2 \cdot \frac{1}{2^{3t}} \)[/tex], we follow these steps:
1. Substitute [tex]\( t = 2 \)[/tex] into the function:
[tex]\[ F(2) = 2 \cdot \frac{1}{2^{3 \cdot 2}} \][/tex]
2. Simplify the exponent:
The exponent in the denominator [tex]\( 3 \cdot 2 \)[/tex] simplifies to:
[tex]\[ 3 \cdot 2 = 6 \][/tex]
This means the expression now looks like:
[tex]\[ F(2) = 2 \cdot \frac{1}{2^6} \][/tex]
3. Evaluate the denominator:
Calculate [tex]\( 2^6 \)[/tex]:
[tex]\[ 2^6 = 64 \][/tex]
4. Put it all together:
Now the expression becomes:
[tex]\[ F(2) = 2 \cdot \frac{1}{64} = 2 \cdot \frac{1}{64} = \frac{2}{64} \][/tex]
5. Simplify [tex]\( \frac{2}{64} \)[/tex]:
Simplify [tex]\( \frac{2}{64} \)[/tex] by dividing the numerator and denominator by 2:
[tex]\[ \frac{2}{64} = \frac{1}{32} \][/tex]
Thus, we arrive at the final value of [tex]\( F(2) \)[/tex]:
[tex]\[ \boxed{\frac{1}{32}} \][/tex]
Therefore, the correct answer is:
D. [tex]\( \frac{1}{32} \)[/tex]
1. Substitute [tex]\( t = 2 \)[/tex] into the function:
[tex]\[ F(2) = 2 \cdot \frac{1}{2^{3 \cdot 2}} \][/tex]
2. Simplify the exponent:
The exponent in the denominator [tex]\( 3 \cdot 2 \)[/tex] simplifies to:
[tex]\[ 3 \cdot 2 = 6 \][/tex]
This means the expression now looks like:
[tex]\[ F(2) = 2 \cdot \frac{1}{2^6} \][/tex]
3. Evaluate the denominator:
Calculate [tex]\( 2^6 \)[/tex]:
[tex]\[ 2^6 = 64 \][/tex]
4. Put it all together:
Now the expression becomes:
[tex]\[ F(2) = 2 \cdot \frac{1}{64} = 2 \cdot \frac{1}{64} = \frac{2}{64} \][/tex]
5. Simplify [tex]\( \frac{2}{64} \)[/tex]:
Simplify [tex]\( \frac{2}{64} \)[/tex] by dividing the numerator and denominator by 2:
[tex]\[ \frac{2}{64} = \frac{1}{32} \][/tex]
Thus, we arrive at the final value of [tex]\( F(2) \)[/tex]:
[tex]\[ \boxed{\frac{1}{32}} \][/tex]
Therefore, the correct answer is:
D. [tex]\( \frac{1}{32} \)[/tex]
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