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Answer :
To solve this problem, we need to find an equivalent expression for the given population formula [tex]\( p = 10000(1.04)^{-t} \)[/tex].
The expression [tex]\( (1.04)^{-t} \)[/tex] can be rewritten by understanding the properties of exponents. Specifically, when we have a negative exponent, it indicates the reciprocal of the base term:
[tex]\[
(1.04)^{-t} = \left(\frac{1}{1.04}\right)^t
\][/tex]
Next, let's simplify [tex]\( \frac{1}{1.04} \)[/tex]:
1. Calculate the reciprocal of 1.04:
[tex]\[
\frac{1}{1.04} \approx 0.9615
\][/tex]
Now, we compare the decimal value of [tex]\( \frac{1}{1.04} \)[/tex] with the fractions provided in the choices to find a match.
2. Evaluate the fractions given in the options:
- [tex]\( \frac{1}{25} \approx 0.04 \)[/tex]
- [tex]\( \frac{25}{26} \approx 0.9615 \)[/tex]
- [tex]\( \frac{26}{25} \approx 1.04 \)[/tex]
- [tex]\( \frac{25}{1} = 25 \)[/tex]
The fraction [tex]\( \frac{25}{26} \)[/tex] approximately equals 0.9615, which is very close to the calculated value of [tex]\( \frac{1}{1.04} \)[/tex].
3. Select the equivalent expression:
The correct choice is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
Therefore, the expression [tex]\( 10000\left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original population formula [tex]\( p = 10000(1.04)^{-t} \)[/tex].
The expression [tex]\( (1.04)^{-t} \)[/tex] can be rewritten by understanding the properties of exponents. Specifically, when we have a negative exponent, it indicates the reciprocal of the base term:
[tex]\[
(1.04)^{-t} = \left(\frac{1}{1.04}\right)^t
\][/tex]
Next, let's simplify [tex]\( \frac{1}{1.04} \)[/tex]:
1. Calculate the reciprocal of 1.04:
[tex]\[
\frac{1}{1.04} \approx 0.9615
\][/tex]
Now, we compare the decimal value of [tex]\( \frac{1}{1.04} \)[/tex] with the fractions provided in the choices to find a match.
2. Evaluate the fractions given in the options:
- [tex]\( \frac{1}{25} \approx 0.04 \)[/tex]
- [tex]\( \frac{25}{26} \approx 0.9615 \)[/tex]
- [tex]\( \frac{26}{25} \approx 1.04 \)[/tex]
- [tex]\( \frac{25}{1} = 25 \)[/tex]
The fraction [tex]\( \frac{25}{26} \)[/tex] approximately equals 0.9615, which is very close to the calculated value of [tex]\( \frac{1}{1.04} \)[/tex].
3. Select the equivalent expression:
The correct choice is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
Therefore, the expression [tex]\( 10000\left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original population formula [tex]\( p = 10000(1.04)^{-t} \)[/tex].
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