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Answer :
To solve the problem of finding an equivalent expression for the population equation [tex]\( p = 10000(1.04)^{-t} \)[/tex], let's break it down step-by-step:
1. Understanding the Given Expression:
The expression we have is:
[tex]\[
p = 10000(1.04)^{-t}
\][/tex]
This can be rewritten as:
[tex]\[
p = 10000 \cdot \frac{1}{(1.04)^t}
\][/tex]
So, we need to find an expression that matches [tex]\(\frac{1}{(1.04)^t}\)[/tex].
2. Evaluating the Options:
Let's go through each of the provided options:
- Option 1: [tex]\(p = 10000\left(\frac{1}{25}\right)^t\)[/tex]
This becomes:
[tex]\[
p = 10000 \cdot \left(\frac{1}{25}\right)^t = 10000 \cdot \frac{1}{25^t}
\][/tex]
This is not equivalent to [tex]\(\frac{1}{(1.04)^t}\)[/tex].
- Option 2: [tex]\(p = 10000\left(\frac{25}{26}\right)^t\)[/tex]
This becomes:
[tex]\[
p = 10000 \cdot \left(\frac{25}{26}\right)^t
\][/tex]
This is still not the same as [tex]\(\frac{1}{(1.04)^t}\)[/tex].
- Option 3: [tex]\(p = 10000\left(\frac{26}{25}\right)^t\)[/tex]
This becomes:
[tex]\[
p = 10000 \cdot \left(\frac{26}{25}\right)^t
\][/tex]
If we rewrite this, it can be seen as:
[tex]\[
p = 10000 \cdot \frac{1}{\left(\frac{25}{26}\right)^t}
\][/tex]
Hence, this matches [tex]\(\frac{1}{(1.04)^t}\)[/tex], since [tex]\(1.04\)[/tex] equals [tex]\(\frac{26}{25}\)[/tex].
- Option 4: [tex]\(\rho = 10000\left(\frac{25}{1}\right)^t\)[/tex]
This becomes:
[tex]\[
\rho = 10000 \cdot 25^t
\][/tex]
This is clearly not equivalent to [tex]\(\frac{1}{(1.04)^t}\)[/tex].
3. Conclusion:
After evaluating each option, the third option:
[tex]\[
p = 10000\left(\frac{26}{25}\right)^t
\][/tex]
is equivalent to the given expression.
Therefore, the correct equivalent expression is found in Option 3: [tex]\(p = 10000\left(\frac{26}{25}\right)^t\)[/tex].
1. Understanding the Given Expression:
The expression we have is:
[tex]\[
p = 10000(1.04)^{-t}
\][/tex]
This can be rewritten as:
[tex]\[
p = 10000 \cdot \frac{1}{(1.04)^t}
\][/tex]
So, we need to find an expression that matches [tex]\(\frac{1}{(1.04)^t}\)[/tex].
2. Evaluating the Options:
Let's go through each of the provided options:
- Option 1: [tex]\(p = 10000\left(\frac{1}{25}\right)^t\)[/tex]
This becomes:
[tex]\[
p = 10000 \cdot \left(\frac{1}{25}\right)^t = 10000 \cdot \frac{1}{25^t}
\][/tex]
This is not equivalent to [tex]\(\frac{1}{(1.04)^t}\)[/tex].
- Option 2: [tex]\(p = 10000\left(\frac{25}{26}\right)^t\)[/tex]
This becomes:
[tex]\[
p = 10000 \cdot \left(\frac{25}{26}\right)^t
\][/tex]
This is still not the same as [tex]\(\frac{1}{(1.04)^t}\)[/tex].
- Option 3: [tex]\(p = 10000\left(\frac{26}{25}\right)^t\)[/tex]
This becomes:
[tex]\[
p = 10000 \cdot \left(\frac{26}{25}\right)^t
\][/tex]
If we rewrite this, it can be seen as:
[tex]\[
p = 10000 \cdot \frac{1}{\left(\frac{25}{26}\right)^t}
\][/tex]
Hence, this matches [tex]\(\frac{1}{(1.04)^t}\)[/tex], since [tex]\(1.04\)[/tex] equals [tex]\(\frac{26}{25}\)[/tex].
- Option 4: [tex]\(\rho = 10000\left(\frac{25}{1}\right)^t\)[/tex]
This becomes:
[tex]\[
\rho = 10000 \cdot 25^t
\][/tex]
This is clearly not equivalent to [tex]\(\frac{1}{(1.04)^t}\)[/tex].
3. Conclusion:
After evaluating each option, the third option:
[tex]\[
p = 10000\left(\frac{26}{25}\right)^t
\][/tex]
is equivalent to the given expression.
Therefore, the correct equivalent expression is found in Option 3: [tex]\(p = 10000\left(\frac{26}{25}\right)^t\)[/tex].
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