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Answer :
Sure! Let's break down the solution step by step:
The problem involves the population of a town, represented by the equation:
[tex]\[ p = 10000 \times (1.04)^{-t} \][/tex]
We need to find which of the following expressions is equivalent to this equation:
1. [tex]\( p = 10000 \left(\frac{1}{25}\right)^t \)[/tex]
2. [tex]\( p = 10000 \left(\frac{25}{26}\right)^t \)[/tex]
3. [tex]\( p = 10000 \left(\frac{26}{25}\right)^t \)[/tex]
4. [tex]\( p = 10000 \left(\frac{25}{1}\right)^t \)[/tex]
Let’s start by rewriting the original equation:
[tex]\[ p = 10000 \times (1.04)^{-t} \][/tex]
Since [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex], we can express:
[tex]\[ p = 10000 \times \left(\frac{1}{1.04}\right)^t \][/tex]
Now, we calculate [tex]\(\frac{1}{1.04}\)[/tex], which is approximately [tex]\(0.9615384615\)[/tex].
We're looking for an equivalent expression of the form:
[tex]\[ p = 10000 \times (b)^t \][/tex]
So the expression should have a base [tex]\( b \approx 0.9615384615 \)[/tex].
Now, we compare this result to the bases from the given options:
1. [tex]\( \left(\frac{1}{25}\right)^t \)[/tex] – which is 0.04, not equivalent.
2. [tex]\( \left(\frac{25}{26}\right)^t \)[/tex] – this is approximately 0.9615384615, which matches our calculated value.
3. [tex]\( \left(\frac{26}{25}\right)^t \)[/tex] – which is approximately 1.04, not equivalent.
4. [tex]\( \left(\frac{25}{1}\right)^t \)[/tex] – this is 25, not equivalent.
Thus, the correct equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
This corresponds to option 2.
The problem involves the population of a town, represented by the equation:
[tex]\[ p = 10000 \times (1.04)^{-t} \][/tex]
We need to find which of the following expressions is equivalent to this equation:
1. [tex]\( p = 10000 \left(\frac{1}{25}\right)^t \)[/tex]
2. [tex]\( p = 10000 \left(\frac{25}{26}\right)^t \)[/tex]
3. [tex]\( p = 10000 \left(\frac{26}{25}\right)^t \)[/tex]
4. [tex]\( p = 10000 \left(\frac{25}{1}\right)^t \)[/tex]
Let’s start by rewriting the original equation:
[tex]\[ p = 10000 \times (1.04)^{-t} \][/tex]
Since [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex], we can express:
[tex]\[ p = 10000 \times \left(\frac{1}{1.04}\right)^t \][/tex]
Now, we calculate [tex]\(\frac{1}{1.04}\)[/tex], which is approximately [tex]\(0.9615384615\)[/tex].
We're looking for an equivalent expression of the form:
[tex]\[ p = 10000 \times (b)^t \][/tex]
So the expression should have a base [tex]\( b \approx 0.9615384615 \)[/tex].
Now, we compare this result to the bases from the given options:
1. [tex]\( \left(\frac{1}{25}\right)^t \)[/tex] – which is 0.04, not equivalent.
2. [tex]\( \left(\frac{25}{26}\right)^t \)[/tex] – this is approximately 0.9615384615, which matches our calculated value.
3. [tex]\( \left(\frac{26}{25}\right)^t \)[/tex] – which is approximately 1.04, not equivalent.
4. [tex]\( \left(\frac{25}{1}\right)^t \)[/tex] – this is 25, not equivalent.
Thus, the correct equivalent expression is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^t \][/tex]
This corresponds to option 2.
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