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The population, [tex]p[/tex], of a town after [tex]t[/tex] years is represented using the equation [tex]p = 10000(1.04)^{-t}[/tex]. Which of the following is an equivalent expression?

A. [tex]p = 10000\left(\frac{1}{25}\right)^t[/tex]
B. [tex]p = 10000\left(\frac{25}{26}\right)^t[/tex]
C. [tex]p = 10000\left(\frac{26}{25}\right)^t[/tex]
D. [tex]p = 10000\left(\frac{25}{1}\right)^t[/tex]

Answer :

To solve the problem of finding an equivalent expression for the population equation [tex]\( p = 10000(1.04)^{-t} \)[/tex], we need to simplify the expression.

1. Understanding the Original Expression:
The expression [tex]\( p = 10000(1.04)^{-t} \)[/tex] indicates that the population decreases over time because of the negative exponent. We need to find an equivalent form that matches one of the options provided.

2. Simplifying the Expression:
To simplify [tex]\( (1.04)^{-t} \)[/tex], recall that a negative exponent means taking the reciprocal of the base:
[tex]\[
(1.04)^{-t} = \left(\frac{1}{1.04}\right)^{t}
\][/tex]

3. Calculate the Reciprocal of 1.04:
Find the reciprocal of 1.04. If [tex]\( 1.04 = \frac{1.04}{1} \)[/tex], then its reciprocal is:
[tex]\[
\frac{1}{1.04} \approx 0.9615
\][/tex]

4. Finding the Fraction Matching the Options:
We need to match this decimal, [tex]\( 0.9615 \)[/tex], with one of the given fractions:
\- Option 1: [tex]\( \left(\frac{1}{25}\right)^t \)[/tex]
\- Option 2: [tex]\( \left(\frac{25}{26}\right)^t \)[/tex]
\- Option 3: [tex]\( \left(\frac{26}{25}\right)^t \)[/tex]
\- Option 4: [tex]\( \left(\frac{25}{1}\right)^t \)[/tex]

Accordingly, compare:
[tex]\[
\frac{25}{26} = 0.9615
\][/tex]

Therefore, the option [tex]\( p=10000\left(\frac{25}{26}\right)^t \)[/tex] matches the simplified form [tex]\( (1.04)^{-t} \)[/tex].

5. Conclusion:

The equivalent expression for the population equation [tex]\( p = 10000(1.04)^{-t} \)[/tex] is:
[tex]\[
p = 10000\left(\frac{25}{26}\right)^t
\][/tex]

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