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Answer :
To find an equivalent expression for the given population equation [tex]\( p = 10000(1.04)^{-t} \)[/tex], let's break down the equation step by step:
1. Understanding the Original Expression:
The original expression is [tex]\( p = 10000(1.04)^{-t} \)[/tex]. The negative exponent indicates that we can rewrite this expression using the reciprocal of the base. Specifically, [tex]\( (1.04)^{-t} \)[/tex] is the same as [tex]\( \left(\frac{1}{1.04}\right)^{t} \)[/tex].
2. Calculate the Reciprocal:
We need to find the reciprocal of 1.04, which is [tex]\( \frac{1}{1.04} \)[/tex]. Calculating this gives us approximately 0.961538.
3. Simplify the Fraction:
The decimal 0.961538 can be expressed as a fraction. This fraction is equivalent to [tex]\( \frac{25}{26} \)[/tex]. Therefore, [tex]\( \left(\frac{1}{1.04}\right)^{t} \)[/tex] can be rewritten as [tex]\( \left(\frac{25}{26}\right)^{t} \)[/tex].
4. Substitute Back into the Expression:
Replace [tex]\( (1.04)^{-t} \)[/tex] in the original equation with [tex]\( \left(\frac{25}{26}\right)^{t} \)[/tex]. This gives us the equivalent expression:
[tex]\[
p = 10000\left(\frac{25}{26}\right)^{t}
\][/tex]
5. Identify the Correct Choice:
From the list of options given, the equivalent expression is:
[tex]\[
p = 10000\left(\frac{25}{26}\right)^{t}
\][/tex]
So, the correct answer is:
[tex]\[
p = 10000\left(\frac{25}{26}\right)^{t}
\][/tex]
1. Understanding the Original Expression:
The original expression is [tex]\( p = 10000(1.04)^{-t} \)[/tex]. The negative exponent indicates that we can rewrite this expression using the reciprocal of the base. Specifically, [tex]\( (1.04)^{-t} \)[/tex] is the same as [tex]\( \left(\frac{1}{1.04}\right)^{t} \)[/tex].
2. Calculate the Reciprocal:
We need to find the reciprocal of 1.04, which is [tex]\( \frac{1}{1.04} \)[/tex]. Calculating this gives us approximately 0.961538.
3. Simplify the Fraction:
The decimal 0.961538 can be expressed as a fraction. This fraction is equivalent to [tex]\( \frac{25}{26} \)[/tex]. Therefore, [tex]\( \left(\frac{1}{1.04}\right)^{t} \)[/tex] can be rewritten as [tex]\( \left(\frac{25}{26}\right)^{t} \)[/tex].
4. Substitute Back into the Expression:
Replace [tex]\( (1.04)^{-t} \)[/tex] in the original equation with [tex]\( \left(\frac{25}{26}\right)^{t} \)[/tex]. This gives us the equivalent expression:
[tex]\[
p = 10000\left(\frac{25}{26}\right)^{t}
\][/tex]
5. Identify the Correct Choice:
From the list of options given, the equivalent expression is:
[tex]\[
p = 10000\left(\frac{25}{26}\right)^{t}
\][/tex]
So, the correct answer is:
[tex]\[
p = 10000\left(\frac{25}{26}\right)^{t}
\][/tex]
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