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Answer :
Sure! Let's break down the problem step-by-step to understand which expression is equivalent to the given equation for the town's population.
We start with the original equation for the population:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
To find an equivalent expression, we need to rewrite the expression [tex]\((1.04)^{-t}\)[/tex] in a different form.
1. Convert 1.04 to a fraction:
- The number 1.04 can be written as [tex]\(\frac{104}{100}\)[/tex].
- Simplifying [tex]\(\frac{104}{100}\)[/tex], we divide both the numerator and the denominator by 4:
[tex]\[
\frac{104}{100} = \frac{26}{25}
\][/tex]
2. Use the properties of exponents:
- When we have [tex]\((\frac{a}{b})^{-t}\)[/tex], it can be rewritten as [tex]\((\frac{b}{a})^t\)[/tex]. This is because [tex]\((\frac{a}{b})^{-t} = \left(\frac{b}{a}\right)^t\)[/tex].
- Apply this to our fraction:
[tex]\[
(1.04)^{-t} = \left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^t
\][/tex]
3. Rewrite the original equation:
- Substitute [tex]\((1.04)^{-t}\)[/tex] with [tex]\((\frac{25}{26})^t\)[/tex] in the population equation:
[tex]\[
p = 10000 \times \left(\frac{25}{26}\right)^t
\][/tex]
So, the equivalent expression is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
This matches with the choice [tex]\(p=10000\left(\frac{25}{26}\right)^t\)[/tex].
We start with the original equation for the population:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
To find an equivalent expression, we need to rewrite the expression [tex]\((1.04)^{-t}\)[/tex] in a different form.
1. Convert 1.04 to a fraction:
- The number 1.04 can be written as [tex]\(\frac{104}{100}\)[/tex].
- Simplifying [tex]\(\frac{104}{100}\)[/tex], we divide both the numerator and the denominator by 4:
[tex]\[
\frac{104}{100} = \frac{26}{25}
\][/tex]
2. Use the properties of exponents:
- When we have [tex]\((\frac{a}{b})^{-t}\)[/tex], it can be rewritten as [tex]\((\frac{b}{a})^t\)[/tex]. This is because [tex]\((\frac{a}{b})^{-t} = \left(\frac{b}{a}\right)^t\)[/tex].
- Apply this to our fraction:
[tex]\[
(1.04)^{-t} = \left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^t
\][/tex]
3. Rewrite the original equation:
- Substitute [tex]\((1.04)^{-t}\)[/tex] with [tex]\((\frac{25}{26})^t\)[/tex] in the population equation:
[tex]\[
p = 10000 \times \left(\frac{25}{26}\right)^t
\][/tex]
So, the equivalent expression is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
This matches with the choice [tex]\(p=10000\left(\frac{25}{26}\right)^t\)[/tex].
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