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Answer :
To determine an equivalent expression for the population of the town, we start with the given equation:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We need to find an expression that has the same result. Let's look at the base of the exponent, [tex]\(1.04^{-t}\)[/tex]:
1. Understanding the negative exponent: A negative exponent, such as [tex]\((1.04)^{-t}\)[/tex], means we take the reciprocal of the base. So, [tex]\((1.04)^{-1}\)[/tex] is the same as [tex]\(\frac{1}{1.04}\)[/tex].
2. Simplifying the reciprocal:
[tex]\[
\frac{1}{1.04} = \frac{1}{\frac{104}{100}} = \frac{100}{104}
\][/tex]
3. Simplifying the fraction:
- Divide the numerator and the denominator by their greatest common divisor, which is 4:
[tex]\[
\frac{100}{104} = \frac{25}{26}
\][/tex]
4. Writing the equivalent expression: Now, replacing [tex]\((1.04)^{-t}\)[/tex] with [tex]\((\frac{25}{26})^{t}\)[/tex], you get:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
This expression is equivalent to the original equation and matches one of the given options. Thus, the correct equivalent expression for the population is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We need to find an expression that has the same result. Let's look at the base of the exponent, [tex]\(1.04^{-t}\)[/tex]:
1. Understanding the negative exponent: A negative exponent, such as [tex]\((1.04)^{-t}\)[/tex], means we take the reciprocal of the base. So, [tex]\((1.04)^{-1}\)[/tex] is the same as [tex]\(\frac{1}{1.04}\)[/tex].
2. Simplifying the reciprocal:
[tex]\[
\frac{1}{1.04} = \frac{1}{\frac{104}{100}} = \frac{100}{104}
\][/tex]
3. Simplifying the fraction:
- Divide the numerator and the denominator by their greatest common divisor, which is 4:
[tex]\[
\frac{100}{104} = \frac{25}{26}
\][/tex]
4. Writing the equivalent expression: Now, replacing [tex]\((1.04)^{-t}\)[/tex] with [tex]\((\frac{25}{26})^{t}\)[/tex], you get:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
This expression is equivalent to the original equation and matches one of the given options. Thus, the correct equivalent expression for the population is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
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