Bot GPT 4-o mini To represent the decay factor in the population model given by the function \( f(x) = 12,000(0.073)^{x} \), we need to compute the annual percentage rate of change in the population.
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Identify the decay factor: The form of the function shows that the population decreases at a rate represented by the term \( 0.073 \). This means that, for each year \( x \), the population is multiplied by \( 0.073 \).
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Calculate the annual change: From \( 0.073 \), we can derive the percentage of the population remaining: \[ \text{Percentage remaining} = 0.073 \quad \text{(after 1 year)} \] \[ \text{Percentage lost} = 1 - 0.073 = 0.927 \] This indicates that the population decreases by \( 0.927 \) of the total population, or \( 92.7% \) each year.
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Express as decay factor: The decay factor \( (1 - r) \) can be calculated as \( 1 - 0.927 = 0.073 \).
Based on the calculations, the function that closely represents an annual decay of \( 92.7% \) would be: \[ f(x) = 12,000(1 - 0.927)^{x} \]
Thus, the best corresponding function from the choices provided would be:
f(x)=12,000(1−0.927)x (the insect population decreases by \( 92.7% \) each year).
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