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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 :

We start with the population function

[tex]$$
p = 10000(1.04)^{-t}.
$$[/tex]

A negative exponent indicates the reciprocal, so we can rewrite the expression as

[tex]$$
p = 10000\left(\frac{1}{1.04}\right)^t.
$$[/tex]

Notice that

[tex]$$
1.04 = \frac{26}{25},
$$[/tex]

so taking the reciprocal gives

[tex]$$
\frac{1}{1.04} = \frac{1}{\frac{26}{25}} = \frac{25}{26}.
$$[/tex]

Therefore, the population function can be rewritten as

[tex]$$
p = 10000\left(\frac{25}{26}\right)^t.
$$[/tex]

Thus, the equivalent expression is

[tex]$$
p = 10000\left(\frac{25}{26}\right)^t.
$$[/tex]

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