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Answer :
We are given the expression
[tex]$$
\rho = 10000\,(1.04)^{-t}.
$$[/tex]
A property of exponents tells us that
[tex]$$
(1.04)^{-t} = \left(\frac{1}{1.04}\right)^t.
$$[/tex]
Since
[tex]$$
1.04 = \frac{26}{25},
$$[/tex]
we can substitute to obtain
[tex]$$
\frac{1}{1.04} = \frac{1}{\frac{26}{25}} = \frac{25}{26}.
$$[/tex]
Thus, the original expression can be rewritten as
[tex]$$
\rho = 10000\,\left(\frac{25}{26}\right)^t.
$$[/tex]
Now, let us compare this with the given multiple-choice options:
1. [tex]$$ p=10000\left(\frac{1}{25}\right)^t $$[/tex]
2. [tex]$$ p=10000\left(\frac{25}{25}\right)^t $$[/tex]
3. [tex]$$ p=10000\left(\frac{26}{25}\right)^t $$[/tex]
4. [tex]$$ p=10000\left(\frac{25}{1}\right)^t $$[/tex]
Notice that the expression we derived is
[tex]$$
10000\left(\frac{25}{26}\right)^t,
$$[/tex]
which is different from each of the four options:
- Option 1 uses [tex]$\frac{1}{25}$[/tex],
- Option 2 uses [tex]$1$[/tex] (since [tex]$\frac{25}{25}=1$[/tex]),
- Option 3 uses [tex]$\frac{26}{25}$[/tex] (which is the reciprocal of [tex]$\frac{25}{26}$[/tex]), and
- Option 4 uses [tex]$25$[/tex].
To confirm further, consider evaluating the expressions for [tex]$t=1$[/tex]:
- The original expression gives
[tex]$$
10000 \left(\frac{25}{26}\right)^1 \approx 10000 \times 0.9615 \approx 9615.38.
$$[/tex]
- Option 1 gives
[tex]$$
10000 \left(\frac{1}{25}\right) \approx 10000 \times 0.04 = 400.
$$[/tex]
- Option 2 gives
[tex]$$
10000 \times 1^1 = 10000.
$$[/tex]
- Option 3 gives
[tex]$$
10000 \left(\frac{26}{25}\right) \approx 10000 \times 1.04 = 10400.
$$[/tex]
- Option 4 gives
[tex]$$
10000 \times 25 = 250000.
$$[/tex]
None of these yield the value [tex]$\approx 9615.38$[/tex].
Therefore, none of the provided choices is equivalent to the original expression [tex]$\rho = 10000\,(1.04)^{-t}$[/tex].
[tex]$$
\rho = 10000\,(1.04)^{-t}.
$$[/tex]
A property of exponents tells us that
[tex]$$
(1.04)^{-t} = \left(\frac{1}{1.04}\right)^t.
$$[/tex]
Since
[tex]$$
1.04 = \frac{26}{25},
$$[/tex]
we can substitute to obtain
[tex]$$
\frac{1}{1.04} = \frac{1}{\frac{26}{25}} = \frac{25}{26}.
$$[/tex]
Thus, the original expression can be rewritten as
[tex]$$
\rho = 10000\,\left(\frac{25}{26}\right)^t.
$$[/tex]
Now, let us compare this with the given multiple-choice options:
1. [tex]$$ p=10000\left(\frac{1}{25}\right)^t $$[/tex]
2. [tex]$$ p=10000\left(\frac{25}{25}\right)^t $$[/tex]
3. [tex]$$ p=10000\left(\frac{26}{25}\right)^t $$[/tex]
4. [tex]$$ p=10000\left(\frac{25}{1}\right)^t $$[/tex]
Notice that the expression we derived is
[tex]$$
10000\left(\frac{25}{26}\right)^t,
$$[/tex]
which is different from each of the four options:
- Option 1 uses [tex]$\frac{1}{25}$[/tex],
- Option 2 uses [tex]$1$[/tex] (since [tex]$\frac{25}{25}=1$[/tex]),
- Option 3 uses [tex]$\frac{26}{25}$[/tex] (which is the reciprocal of [tex]$\frac{25}{26}$[/tex]), and
- Option 4 uses [tex]$25$[/tex].
To confirm further, consider evaluating the expressions for [tex]$t=1$[/tex]:
- The original expression gives
[tex]$$
10000 \left(\frac{25}{26}\right)^1 \approx 10000 \times 0.9615 \approx 9615.38.
$$[/tex]
- Option 1 gives
[tex]$$
10000 \left(\frac{1}{25}\right) \approx 10000 \times 0.04 = 400.
$$[/tex]
- Option 2 gives
[tex]$$
10000 \times 1^1 = 10000.
$$[/tex]
- Option 3 gives
[tex]$$
10000 \left(\frac{26}{25}\right) \approx 10000 \times 1.04 = 10400.
$$[/tex]
- Option 4 gives
[tex]$$
10000 \times 25 = 250000.
$$[/tex]
None of these yield the value [tex]$\approx 9615.38$[/tex].
Therefore, none of the provided choices is equivalent to the original expression [tex]$\rho = 10000\,(1.04)^{-t}$[/tex].
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