Question

The change in the population, P, of a colony of insects for x years is represented by the function f(x)=12,000(0.073)x52
.

Select the function whose decay factor best represents the annual percentage rate of change for the insect population?

(1 point)
Responses

f(x)=12,000(1−0.951)x
f of x is equal to 12 comma 000 times open paren 1 minus 0 point 9 5 1 close paren to the x th power

f(x)=12,000(1−0.9986)x
f of x is equal to 12 comma 000 times open paren 1 minus 0 point 9 9 8 6 close paren to the x th power

f(x)=12,000(1−0.979)x
f of x is equal to 12 comma 000 times open paren 1 minus 0 point 9 7 9 close paren to the x th power

f(x)=12,000(1−0.927)x
f of x is equal to 12 comma 000 times open paren 1 minus 0 point 9 2 7 close paren to the x th power

Answer 1

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.

1. **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 \( 0.073 \).

2. **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.

3. **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).