Answer :
To solve the problem of finding an equivalent expression for the given population equation [tex]\( p = 10000(1.04)^{-t} \)[/tex], we need to simplify and transform the expression inside the parentheses, [tex]\( (1.04)^{-t} \)[/tex].
1. Understanding the expression [tex]\((1.04)^{-t}\)[/tex]:
- The expression [tex]\( (1.04)^{-t} \)[/tex] is equivalent to [tex]\( \left(\frac{1}{1.04}\right)^t \)[/tex]. This transformation uses the rule [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex].
2. Simplifying [tex]\(\frac{1}{1.04}\)[/tex]:
- When we calculate [tex]\( \frac{1}{1.04} \)[/tex], we get approximately 0.9615 as a decimal.
3. Expressing 0.9615 as a fraction:
- The decimal 0.9615 is equivalent to the fraction [tex]\(\frac{25}{26}\)[/tex].
- Thus, we can express [tex]\( (1.04)^{-t} \)[/tex] as [tex]\( \left(\frac{25}{26}\right)^t \)[/tex].
4. Forming the equivalent expression:
- Therefore, the entire population expression can be rewritten as:
[tex]\[
p = 10000 \left( \frac{25}{26} \right)^t
\][/tex]
Upon examining the options provided:
- The third option matches our rewritten expression:
[tex]\[
p = 10000 \left( \frac{26}{25} \right)^t
\][/tex]
However, notice that there seems to be an error in the original working, as it should actually be [tex]\( p = 10000 \left(\frac{25}{26}\right)^t \)[/tex]. It appears the intended answer should reflect this equivalence.
Therefore, the correct equivalent expression for the equation is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
But since the choices seem to involve flipping this relationship, normally, this might imply a typo or misunderstanding in the options, as technically [tex]\( \frac{26}{25} \)[/tex] involves a growth factor, not a decay factor like [tex]\(\frac{25}{26}\)[/tex]. However, we identified it as [tex]\(\frac{25}{26}\)[/tex] due to the calculation shown.
1. Understanding the expression [tex]\((1.04)^{-t}\)[/tex]:
- The expression [tex]\( (1.04)^{-t} \)[/tex] is equivalent to [tex]\( \left(\frac{1}{1.04}\right)^t \)[/tex]. This transformation uses the rule [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex].
2. Simplifying [tex]\(\frac{1}{1.04}\)[/tex]:
- When we calculate [tex]\( \frac{1}{1.04} \)[/tex], we get approximately 0.9615 as a decimal.
3. Expressing 0.9615 as a fraction:
- The decimal 0.9615 is equivalent to the fraction [tex]\(\frac{25}{26}\)[/tex].
- Thus, we can express [tex]\( (1.04)^{-t} \)[/tex] as [tex]\( \left(\frac{25}{26}\right)^t \)[/tex].
4. Forming the equivalent expression:
- Therefore, the entire population expression can be rewritten as:
[tex]\[
p = 10000 \left( \frac{25}{26} \right)^t
\][/tex]
Upon examining the options provided:
- The third option matches our rewritten expression:
[tex]\[
p = 10000 \left( \frac{26}{25} \right)^t
\][/tex]
However, notice that there seems to be an error in the original working, as it should actually be [tex]\( p = 10000 \left(\frac{25}{26}\right)^t \)[/tex]. It appears the intended answer should reflect this equivalence.
Therefore, the correct equivalent expression for the equation is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
But since the choices seem to involve flipping this relationship, normally, this might imply a typo or misunderstanding in the options, as technically [tex]\( \frac{26}{25} \)[/tex] involves a growth factor, not a decay factor like [tex]\(\frac{25}{26}\)[/tex]. However, we identified it as [tex]\(\frac{25}{26}\)[/tex] due to the calculation shown.