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Answer :
Sure! Let's go through the process of finding the equivalent expression for the given population equation:
The original equation for the population, [tex]\( p \)[/tex], of a town after [tex]\( t \)[/tex] years is given by:
[tex]\[ p = 10000 \times (1.04)^{-t} \][/tex]
We need to find an equivalent expression using the given choices. Let's break it down:
1. Express the Growth Factor as a Fraction:
The growth factor here is [tex]\( 1.04 \)[/tex]. We can express [tex]\( 1.04 \)[/tex] as a fraction:
[tex]\[
1.04 = \frac{104}{100} = \frac{26}{25}
\][/tex]
2. Understand the Negative Exponent:
The equation given is [tex]\( (1.04)^{-t} \)[/tex]. Using the fraction we found:
[tex]\[
(1.04)^{-t} = \left(\frac{26}{25}\right)^{-t}
\][/tex]
3. Apply the Negative Exponent Rule:
A negative exponent means we take the reciprocal of the base and make the exponent positive:
[tex]\[
\left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^{t}
\][/tex]
4. Substitute Back into the Original Equation:
Now, substitute the fraction back into the equation:
[tex]\[
p = 10000 \times \left(\frac{25}{26}\right)^{t}
\][/tex]
So, the equivalent expression for the population of the town in that form is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^{t} \][/tex]
This matches option: [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex].
The original equation for the population, [tex]\( p \)[/tex], of a town after [tex]\( t \)[/tex] years is given by:
[tex]\[ p = 10000 \times (1.04)^{-t} \][/tex]
We need to find an equivalent expression using the given choices. Let's break it down:
1. Express the Growth Factor as a Fraction:
The growth factor here is [tex]\( 1.04 \)[/tex]. We can express [tex]\( 1.04 \)[/tex] as a fraction:
[tex]\[
1.04 = \frac{104}{100} = \frac{26}{25}
\][/tex]
2. Understand the Negative Exponent:
The equation given is [tex]\( (1.04)^{-t} \)[/tex]. Using the fraction we found:
[tex]\[
(1.04)^{-t} = \left(\frac{26}{25}\right)^{-t}
\][/tex]
3. Apply the Negative Exponent Rule:
A negative exponent means we take the reciprocal of the base and make the exponent positive:
[tex]\[
\left(\frac{26}{25}\right)^{-t} = \left(\frac{25}{26}\right)^{t}
\][/tex]
4. Substitute Back into the Original Equation:
Now, substitute the fraction back into the equation:
[tex]\[
p = 10000 \times \left(\frac{25}{26}\right)^{t}
\][/tex]
So, the equivalent expression for the population of the town in that form is:
[tex]\[ p = 10000 \left(\frac{25}{26}\right)^{t} \][/tex]
This matches option: [tex]\( p = 10000\left(\frac{25}{26}\right)^t \)[/tex].
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