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Answer :
Sure, let's break down the problem step by step.
The original equation given for the population [tex]\( p \)[/tex] of the town after [tex]\( t \)[/tex] years is:
[tex]\[ p = 10000 \times (104)^{-t} \][/tex]
Our goal is to identify an equivalent expression from the given options. Let's understand the transformation step by step:
1. Understanding the base and exponent:
- The expression [tex]\((104)^{-t}\)[/tex] means we can take the reciprocal of 104 and raise it to the power of [tex]\( t \)[/tex]. This is based on the property of exponents: [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex].
- Therefore, [tex]\((104)^{-t} = \left(\frac{1}{104}\right)^t\)[/tex].
2. Look at the equivalent expressions:
- From the above transformation, [tex]\( \left(\frac{1}{104}\right)^t \)[/tex] is the rewritten expression.
- We need to match [tex]\(\left(\frac{1}{104}\right)^t\)[/tex] with one of the given options.
3. Comparing with provided options:
- The expression [tex]\(\left(\frac{1}{104}\right)\)[/tex] as a decimal is approximately 0.009615384615384616 (calculated by dividing 1 by 104).
- We need to find which of the provided options matches this base when expressed as a decimal.
4. Verify with options:
- Let's check the options:
- Option 1: [tex]\(\left(\frac{1}{25}\right)^t\)[/tex] which is 0.04, does not match.
- Option 2: [tex]\(\left(\frac{25}{26}\right)^t\)[/tex] evaluates to about 0.9615, this matches the reciprocal idea, as when divided [tex]\( \frac{25}{26} = 0.9615\)[/tex], this is the correct match.
- Option 3 and 4 do not apply based on their fractions.
So, the expression [tex]\( p = 10000 \times \left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original expression.
The original equation given for the population [tex]\( p \)[/tex] of the town after [tex]\( t \)[/tex] years is:
[tex]\[ p = 10000 \times (104)^{-t} \][/tex]
Our goal is to identify an equivalent expression from the given options. Let's understand the transformation step by step:
1. Understanding the base and exponent:
- The expression [tex]\((104)^{-t}\)[/tex] means we can take the reciprocal of 104 and raise it to the power of [tex]\( t \)[/tex]. This is based on the property of exponents: [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex].
- Therefore, [tex]\((104)^{-t} = \left(\frac{1}{104}\right)^t\)[/tex].
2. Look at the equivalent expressions:
- From the above transformation, [tex]\( \left(\frac{1}{104}\right)^t \)[/tex] is the rewritten expression.
- We need to match [tex]\(\left(\frac{1}{104}\right)^t\)[/tex] with one of the given options.
3. Comparing with provided options:
- The expression [tex]\(\left(\frac{1}{104}\right)\)[/tex] as a decimal is approximately 0.009615384615384616 (calculated by dividing 1 by 104).
- We need to find which of the provided options matches this base when expressed as a decimal.
4. Verify with options:
- Let's check the options:
- Option 1: [tex]\(\left(\frac{1}{25}\right)^t\)[/tex] which is 0.04, does not match.
- Option 2: [tex]\(\left(\frac{25}{26}\right)^t\)[/tex] evaluates to about 0.9615, this matches the reciprocal idea, as when divided [tex]\( \frac{25}{26} = 0.9615\)[/tex], this is the correct match.
- Option 3 and 4 do not apply based on their fractions.
So, the expression [tex]\( p = 10000 \times \left(\frac{25}{26}\right)^t \)[/tex] is equivalent to the original expression.
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