Answer
a. growth model is $ A= 6.04e^{0.01\cdot t}$
b. in $2040$
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.
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$ a.$
Here, we take $ t=0$ in the year 2000. We are given $ A_{0}=6.04$ (million).
50 years later, when $ t=50$, we expect $ A=10.0$ (million)
We solve for k in
$ 10.0=6.04e^{k\cdot 50}\qquad $... /$\div 6.04$
$ 1.6656=e^{k\cdot 50}\qquad $... $/\ln(...)$
$ 0.50418=50k\qquad $... /$\div 50$
$ k\approx 0.01$
So, our growth model is $ A= 6.04e^{0.01\cdot t}$
$ b.$
Solve for t when $ A=9.0$
$ 9.0=6.04e^{0.01\cdot t}\qquad $... /$\div 6.04$
$ 1.49\approx e^{0.01t}\qquad $... $/\ln(...)$
$ 0.3988\approx 0.01t\qquad $... $/\times 100$
$ t\approx 39.88\approx 40$
It will take about 40 years (after 2000), so the answer is
in $2040$.
