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

Question

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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(1−0.9986)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(1−0.927)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

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(1−0.979)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
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Answer 1

To determine which function best represents the annual percentage rate of change for the insect population described by the function \( f(x) = 12,000(0.073)x^{52} \), we need to analyze the decay factor present in this equation.

The general exponential decay function can be expressed as: \[ P(t) = P_0 \times (1 - r)^t \] where \( r \) is the decay rate.

First, we need to find the decay factor from the given function \( f(x) = 12,000(0.073)x^{52} \). However, it seems that the function is already written in terms of growth rather than decay, and it uses an exponent \( 52 \) which may imply a different structure.

To find the annual percentage change, we take the decay factor, which is \( 0.073 \), and express it in terms of an annual percentage by converting it to the form \( (1 - r) \).

To do this for the decay factor:

1. The term \( 0.073 \) suggests a small growth or decay factor.
2. If we subtract \( 0.073 \) from \( 1 \) to find \( r \): \[ r = 1 - 0.073 = 0.927 \]

Conclusion

This indicates that the annual decay rate (\( r \)) is \( 0.927 \). Therefore, our decay factor can be expressed as \( 1 - r = 1 - 0.927 \).

Hence, the corresponding function from the choices given will be: \[ f(x) = 12,000(1 - 0.927)^x \]

So, the correct response is: f(x) = 12,000(1 - 0.927)x