We have the function
![p(t)=550(1-e^{-0.039t})](https://tex.z-dn.net/?f=p%28t%29%3D550%281-e%5E%7B-0.039t%7D%29)
Therefore we want to determine when we have
![p(t_0)=550](https://tex.z-dn.net/?f=p%28t_0%29%3D550)
It means that the term
![e^{-0.039t}](https://tex.z-dn.net/?f=e%5E%7B-0.039t%7D)
Must go to zero, then let's forget the rest of the function for a sec and focus only on this term
![e^{-0.039t}\rightarrow0](https://tex.z-dn.net/?f=e%5E%7B-0.039t%7D%5Crightarrow0)
But for which value of t? When we have a decreasing exponential, it's interesting to input values that are multiples of the exponential coefficient, if we have 0.039 in the exponential, let's define that
![\alpha=\frac{1}{0.039}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cfrac%7B1%7D%7B0.039%7D)
The inverse of the number, but why do that? look what happens when we do t = α
![e^{-0.039t}\Rightarrow e^{-0.039\alpha}\Rightarrow e^{-1}=\frac{1}{e}](https://tex.z-dn.net/?f=e%5E%7B-0.039t%7D%5CRightarrow%20e%5E%7B-0.039%5Calpha%7D%5CRightarrow%20e%5E%7B-1%7D%3D%5Cfrac%7B1%7D%7Be%7D)
And when t = 2α
![e^{-0.039t}\Rightarrow e^{-0.039\cdot2\alpha}\Rightarrow e^{-2}=\frac{1}{e^2}](https://tex.z-dn.net/?f=e%5E%7B-0.039t%7D%5CRightarrow%20e%5E%7B-0.039%5Ccdot2%5Calpha%7D%5CRightarrow%20e%5E%7B-2%7D%3D%5Cfrac%7B1%7D%7Be%5E2%7D)
We can write it in terms of e only.
And we can find for which value of α we have a small value that satisfies
![e^{-0.039t}\approx0](https://tex.z-dn.net/?f=e%5E%7B-0.039t%7D%5Capprox0)
Only using powers of e
Let's write some inverse powers of e:
![\begin{gathered} \frac{1}{e}=0.368 \\ \\ \frac{1}{e^2}=0.135 \\ \\ \frac{1}{e^3}=0.05 \\ \\ \frac{1}{e^4}=0.02 \\ \\ \frac{1}{e^5}=0.006 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cfrac%7B1%7D%7Be%7D%3D0.368%20%5C%5C%20%20%5C%5C%20%5Cfrac%7B1%7D%7Be%5E2%7D%3D0.135%20%5C%5C%20%20%5C%5C%20%5Cfrac%7B1%7D%7Be%5E3%7D%3D0.05%20%5C%5C%20%20%5C%5C%20%5Cfrac%7B1%7D%7Be%5E4%7D%3D0.02%20%5C%5C%20%20%5C%5C%20%5Cfrac%7B1%7D%7Be%5E5%7D%3D0.006%20%5Cend%7Bgathered%7D)
See that at t = 5α we have a small value already, then if we input p(5α) we can get
![\begin{gathered} p(5\alpha)=550(1-e^{-0.039\cdot5\alpha}) \\ \\ p(5\alpha)=550(1-0.006) \\ \\ p(5\alpha)=550(1-0.006) \\ \\ p(5\alpha)=550\cdot0.994 \\ \\ p(5\alpha)\approx547 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20p%285%5Calpha%29%3D550%281-e%5E%7B-0.039%5Ccdot5%5Calpha%7D%29%20%5C%5C%20%20%5C%5C%20p%285%5Calpha%29%3D550%281-0.006%29%20%5C%5C%20%20%5C%5C%20p%285%5Calpha%29%3D550%281-0.006%29%20%5C%5C%20%20%5C%5C%20p%285%5Calpha%29%3D550%5Ccdot0.994%20%5C%5C%20%20%5C%5C%20p%285%5Calpha%29%5Capprox547%20%5Cend%7Bgathered%7D)
That's already very close to 550, if we want a better approximation we can use t = 8α, which will result in 549.81, which is basically 550.
Therefore, we can use t = 5α and say that 3 people are not important for our case, and say that it's basically 550, or use t = 8α and get a very close value.
In both cases, the decimal answers would be