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Answer :
To determine which expression is equivalent to the given equation [tex]\( p = 10000(1.04)^{-t} \)[/tex], we need to find a base of the form [tex]\( \frac{a}{b} \)[/tex] that, when raised to the power of [tex]\(-t\)[/tex], matches [tex]\( (1.04)^{-t} \)[/tex].
The task is to identify an expression where:
[tex]\[ (1.04)^{-t} = \left(\frac{a}{b}\right)^t \][/tex]
This means:
[tex]\[ \left(\frac{1}{1.04}\right)^t = \left(\frac{a}{b}\right)^t \][/tex]
Therefore, we need to calculate [tex]\( \frac{1}{1.04} \)[/tex].
Calculating [tex]\( \frac{1}{1.04} \)[/tex] gives us approximately [tex]\( 0.96154 \)[/tex].
Now, let's compare this value with the options given:
- Option 1: [tex]\( \left(\frac{1}{25}\right)^t = 0.04 \)[/tex]
- Option 2: [tex]\( \left(\frac{25}{26}\right)^t \approx 0.96154 \)[/tex]
- Option 3: [tex]\( \left(\frac{26}{25}\right)^t \approx 1.04 \)[/tex]
- Option 4: [tex]\( \left(\frac{25}{1}\right)^t = 25 \)[/tex]
Matching calculated value [tex]\( 0.96154 \)[/tex] to the options above, we find that:
The equivalent expression is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
This means the correct answer is the second option: [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex].
The task is to identify an expression where:
[tex]\[ (1.04)^{-t} = \left(\frac{a}{b}\right)^t \][/tex]
This means:
[tex]\[ \left(\frac{1}{1.04}\right)^t = \left(\frac{a}{b}\right)^t \][/tex]
Therefore, we need to calculate [tex]\( \frac{1}{1.04} \)[/tex].
Calculating [tex]\( \frac{1}{1.04} \)[/tex] gives us approximately [tex]\( 0.96154 \)[/tex].
Now, let's compare this value with the options given:
- Option 1: [tex]\( \left(\frac{1}{25}\right)^t = 0.04 \)[/tex]
- Option 2: [tex]\( \left(\frac{25}{26}\right)^t \approx 0.96154 \)[/tex]
- Option 3: [tex]\( \left(\frac{26}{25}\right)^t \approx 1.04 \)[/tex]
- Option 4: [tex]\( \left(\frac{25}{1}\right)^t = 25 \)[/tex]
Matching calculated value [tex]\( 0.96154 \)[/tex] to the options above, we find that:
The equivalent expression is:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
This means the correct answer is the second option: [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex].
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