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Answer :
To solve the problem, we need to determine which of the given options is equivalent to the expression [tex]\( p = 10000(1.04)^{-t} \)[/tex].
Let's break down each option:
1. Option 1: [tex]\( p = 10000\left(\frac{1}{25}\right)^t \)[/tex]
This expression is equivalent to [tex]\( p = 10000 \times (25^{-t}) \)[/tex], which is not the same as [tex]\( p = 10000(1.04)^{-t} \)[/tex].
2. Option 2: [tex]\( p = 10000\left(\frac{25}{26}\right) \)[/tex]
This option is incomplete and incorrect because it does not account for the variable [tex]\( t \)[/tex].
3. Option 3: [tex]\( p = 10000\left(\frac{26}{25}\right)^{\prime} \)[/tex]
This should be read as [tex]\( p = 10000\left(\frac{26}{25}\right)^{-t} \)[/tex] to compare. Note that [tex]\( 1.04 \)[/tex] as a fraction is approximately [tex]\( \frac{26}{25} \)[/tex]. Therefore, [tex]\( (1.04)^{-t} \)[/tex] is equivalent to [tex]\( \left(\frac{26}{25}\right)^{-t} \)[/tex].
4. Option 4: [tex]\( p = 10000\left(\frac{25}{1}\right)^t \)[/tex]
This expression simplifies to [tex]\( p = 10000 \times (25^t) \)[/tex], which is clearly not equivalent to [tex]\( p = 10000(1.04)^{-t} \)[/tex].
From the analysis above, Option 3 is the correct equivalent expression:
[tex]\[ p = 10000\left(\frac{26}{25}\right)^{-t} \][/tex]
Let's break down each option:
1. Option 1: [tex]\( p = 10000\left(\frac{1}{25}\right)^t \)[/tex]
This expression is equivalent to [tex]\( p = 10000 \times (25^{-t}) \)[/tex], which is not the same as [tex]\( p = 10000(1.04)^{-t} \)[/tex].
2. Option 2: [tex]\( p = 10000\left(\frac{25}{26}\right) \)[/tex]
This option is incomplete and incorrect because it does not account for the variable [tex]\( t \)[/tex].
3. Option 3: [tex]\( p = 10000\left(\frac{26}{25}\right)^{\prime} \)[/tex]
This should be read as [tex]\( p = 10000\left(\frac{26}{25}\right)^{-t} \)[/tex] to compare. Note that [tex]\( 1.04 \)[/tex] as a fraction is approximately [tex]\( \frac{26}{25} \)[/tex]. Therefore, [tex]\( (1.04)^{-t} \)[/tex] is equivalent to [tex]\( \left(\frac{26}{25}\right)^{-t} \)[/tex].
4. Option 4: [tex]\( p = 10000\left(\frac{25}{1}\right)^t \)[/tex]
This expression simplifies to [tex]\( p = 10000 \times (25^t) \)[/tex], which is clearly not equivalent to [tex]\( p = 10000(1.04)^{-t} \)[/tex].
From the analysis above, Option 3 is the correct equivalent expression:
[tex]\[ p = 10000\left(\frac{26}{25}\right)^{-t} \][/tex]
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