Answer
$-0.0039$
Work Step by Step
Exponential growth model $: \quad A= A_{0}e^{kt}$.
If $ k \gt 0$, the function models the amount of a growing entity.
$ A_{0}$ is the original amount, or size, of the growing entity at time t = 0,
$ A $ is the amount at time $ t $, and
$ k $ is a constant representing the growth rate.
If k is negative, the model is an exponential decay model.
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Model: $ A=A_{0}e^{kt}$
Unknown: $ k $, when $ A_{0}=82.3, A=70.5, t=40,$
$ 70.5=82.3e^{k\cdot 40}\qquad $... /$\div 82.3$
$ 0.85662\approx e^{k\cdot 40}\qquad $... $/\ln(...)$
$-0.14576=40k\qquad $... /$\div 40$
$ k\approx-0.0039$
