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Answer :
To solve this problem, we need to determine which of the given expressions is equivalent to the original expression for the population: [tex]\( p = 10000(1.04)^{-t} \)[/tex].
Here is a step-by-step approach to understanding the equivalent expressions:
1. Analyze the original expression:
- The original expression is [tex]\( 1.04^{-t} \)[/tex].
- This can be rewritten with a base change to identify its equivalent form.
2. Rewriting exponential expressions:
- The expression [tex]\( 1.04^{-t} \)[/tex] can be rewritten using the property of exponents: [tex]\( a^{-n} = \left(\frac{1}{a}\right)^{n} \)[/tex].
- So, [tex]\( 1.04^{-t} = \left(\frac{1}{1.04}\right)^{t} \)[/tex].
3. Find the equivalent fraction:
- Calculate [tex]\( \frac{1}{1.04} \)[/tex].
- This fraction simplifies to approximately [tex]\( \frac{25}{26} \)[/tex].
4. Match the options with the rewritten form:
- Option 1: [tex]\( \left(\frac{1}{25}\right)^t \)[/tex] – does not match.
- Option 2: [tex]\( \left(\frac{25}{26}\right)^t \)[/tex] – this matches because it represents the same change in the base as calculated.
- Option 3: [tex]\( \left(\frac{26}{25}\right)^t \)[/tex] – does not match.
- Option 4: [tex]\( \left(\frac{25}{1}\right)^t \)[/tex] – does not match.
Therefore, the expression [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original population model [tex]\( p = 10000(1.04)^{-t} \)[/tex]. Hence, the correct choice is the second option:
[tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex].
Here is a step-by-step approach to understanding the equivalent expressions:
1. Analyze the original expression:
- The original expression is [tex]\( 1.04^{-t} \)[/tex].
- This can be rewritten with a base change to identify its equivalent form.
2. Rewriting exponential expressions:
- The expression [tex]\( 1.04^{-t} \)[/tex] can be rewritten using the property of exponents: [tex]\( a^{-n} = \left(\frac{1}{a}\right)^{n} \)[/tex].
- So, [tex]\( 1.04^{-t} = \left(\frac{1}{1.04}\right)^{t} \)[/tex].
3. Find the equivalent fraction:
- Calculate [tex]\( \frac{1}{1.04} \)[/tex].
- This fraction simplifies to approximately [tex]\( \frac{25}{26} \)[/tex].
4. Match the options with the rewritten form:
- Option 1: [tex]\( \left(\frac{1}{25}\right)^t \)[/tex] – does not match.
- Option 2: [tex]\( \left(\frac{25}{26}\right)^t \)[/tex] – this matches because it represents the same change in the base as calculated.
- Option 3: [tex]\( \left(\frac{26}{25}\right)^t \)[/tex] – does not match.
- Option 4: [tex]\( \left(\frac{25}{1}\right)^t \)[/tex] – does not match.
Therefore, the expression [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original population model [tex]\( p = 10000(1.04)^{-t} \)[/tex]. Hence, the correct choice is the second option:
[tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex].
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