High School

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The population, [tex] p [/tex], of a town after [tex] t [/tex] years is represented by 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 equation

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

Because the exponent is negative, we can rewrite the expression as

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

Next, notice that

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

Taking the reciprocal of [tex]$\frac{26}{25}$[/tex] yields

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

Substituting this back into our equation for [tex]$p$[/tex], we have

[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]

which corresponds to the second option.

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