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Answer :
Sure! Let's find the equivalent expression step-by-step.
The original equation given for the population, [tex]\( p \)[/tex], is:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
The problem is asking us to find an equivalent expression. We need to rewrite [tex]\( (1.04)^{-t} \)[/tex] in a different form.
### Steps to Solve:
1. Understanding Negative Exponents:
The expression [tex]\( (1.04)^{-t} \)[/tex] can be rewritten because a negative exponent means taking the reciprocal. Therefore:
[tex]\[ (1.04)^{-t} = \left(\frac{1}{1.04}\right)^{t} \][/tex]
2. Convert 1.04 to a Fraction:
The decimal 1.04 can be written as a fraction:
[tex]\[ 1.04 = \frac{26}{25} \][/tex]
3. Finding the Reciprocal of the Fraction:
Now we find the reciprocal of [tex]\( \frac{26}{25} \)[/tex]:
[tex]\[ \frac{1}{1.04} = \frac{1}{\frac{26}{25}} = \frac{25}{26} \][/tex]
4. Substitute Back into the Expression:
Replace [tex]\( \left(\frac{1}{1.04}\right)^{t} \)[/tex] with [tex]\( \left(\frac{25}{26}\right)^{t} \)[/tex]:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^{t} \][/tex]
### Conclusion:
The equivalent expression to [tex]\( p = 10000(1.04)^{-t} \)[/tex] is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^{t} \][/tex]
So, the correct option is:
[tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex]
This matches the second option in the list provided.
The original equation given for the population, [tex]\( p \)[/tex], is:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
The problem is asking us to find an equivalent expression. We need to rewrite [tex]\( (1.04)^{-t} \)[/tex] in a different form.
### Steps to Solve:
1. Understanding Negative Exponents:
The expression [tex]\( (1.04)^{-t} \)[/tex] can be rewritten because a negative exponent means taking the reciprocal. Therefore:
[tex]\[ (1.04)^{-t} = \left(\frac{1}{1.04}\right)^{t} \][/tex]
2. Convert 1.04 to a Fraction:
The decimal 1.04 can be written as a fraction:
[tex]\[ 1.04 = \frac{26}{25} \][/tex]
3. Finding the Reciprocal of the Fraction:
Now we find the reciprocal of [tex]\( \frac{26}{25} \)[/tex]:
[tex]\[ \frac{1}{1.04} = \frac{1}{\frac{26}{25}} = \frac{25}{26} \][/tex]
4. Substitute Back into the Expression:
Replace [tex]\( \left(\frac{1}{1.04}\right)^{t} \)[/tex] with [tex]\( \left(\frac{25}{26}\right)^{t} \)[/tex]:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^{t} \][/tex]
### Conclusion:
The equivalent expression to [tex]\( p = 10000(1.04)^{-t} \)[/tex] is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^{t} \][/tex]
So, the correct option is:
[tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex]
This matches the second option in the list provided.
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