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Answer :
Sure! Let's solve the problem step-by-step.
We are given the population function:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We need to find an equivalent expression from the provided options.
First, let's understand what the given expression means. The base [tex]\( 1.04 \)[/tex] raised to the power of [tex]\(-t\)[/tex] can be rewritten using the properties of exponents:
[tex]\[ (1.04)^{-t} = \left(\frac{1}{1.04}\right)^t \][/tex]
Next, we need to simplify [tex]\( \frac{1}{1.04} \)[/tex] to see which of the provided options it matches. To do this, we recognize that:
[tex]\[ 1.04 \approx \frac{26}{25} \][/tex]
Therefore:
[tex]\[ \frac{1}{1.04} \approx \frac{1}{\frac{26}{25}} = \frac{25}{26} \][/tex]
Now substituting this back in:
[tex]\[ \left(\frac{1}{1.04}\right)^t \approx \left(\frac{25}{26}\right)^t \][/tex]
Thus, the expression [tex]\( 10000(1.04)^{-t} \)[/tex] can be equivalently written as:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
So, the correct answer is:
[tex]\[ p=10000\left(\frac{25}{26}\right)^t \][/tex]
We are given the population function:
[tex]\[ p = 10000(1.04)^{-t} \][/tex]
We need to find an equivalent expression from the provided options.
First, let's understand what the given expression means. The base [tex]\( 1.04 \)[/tex] raised to the power of [tex]\(-t\)[/tex] can be rewritten using the properties of exponents:
[tex]\[ (1.04)^{-t} = \left(\frac{1}{1.04}\right)^t \][/tex]
Next, we need to simplify [tex]\( \frac{1}{1.04} \)[/tex] to see which of the provided options it matches. To do this, we recognize that:
[tex]\[ 1.04 \approx \frac{26}{25} \][/tex]
Therefore:
[tex]\[ \frac{1}{1.04} \approx \frac{1}{\frac{26}{25}} = \frac{25}{26} \][/tex]
Now substituting this back in:
[tex]\[ \left(\frac{1}{1.04}\right)^t \approx \left(\frac{25}{26}\right)^t \][/tex]
Thus, the expression [tex]\( 10000(1.04)^{-t} \)[/tex] can be equivalently written as:
[tex]\[ p = 10000\left(\frac{25}{26}\right)^t \][/tex]
So, the correct answer is:
[tex]\[ p=10000\left(\frac{25}{26}\right)^t \][/tex]
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