Answer:
a) ![N(t=10) = \frac{95}{1+8.5 e^{-0.12(10)}}= \frac{95}{1+ 8.5 e^{-1.2}} = 26.684](https://tex.z-dn.net/?f=%20N%28t%3D10%29%20%3D%20%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12%2810%29%7D%7D%3D%20%5Cfrac%7B95%7D%7B1%2B%208.5%20e%5E%7B-1.2%7D%7D%20%3D%2026.684)
b) ![N(t=20) = \frac{95}{1+8.5 e^{-0.12(20)}}= \frac{95}{1+ 8.5 e^{-2.4}} = 53.639](https://tex.z-dn.net/?f=%20N%28t%3D20%29%20%3D%20%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12%2820%29%7D%7D%3D%20%5Cfrac%7B95%7D%7B1%2B%208.5%20e%5E%7B-2.4%7D%7D%20%3D%2053.639)
c) ![70 =\frac{95}{1+8.5 e^{-0.12t}}](https://tex.z-dn.net/?f=%2070%20%3D%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12t%7D%7D)
![1+ 8.5 e^{-0.12t} = \frac{95}{70}= \frac{19}{14}](https://tex.z-dn.net/?f=%201%2B%208.5%20e%5E%7B-0.12t%7D%20%3D%20%5Cfrac%7B95%7D%7B70%7D%3D%20%5Cfrac%7B19%7D%7B14%7D)
![8.5 e^{-0.12t} = \frac{19}{14}-1= \frac{5}{14}](https://tex.z-dn.net/?f=8.5%20e%5E%7B-0.12t%7D%20%3D%20%5Cfrac%7B19%7D%7B14%7D-1%3D%20%5Cfrac%7B5%7D%7B14%7D)
![e^{-0.12t} = \frac{\frac{5}{14}}{8.5}= \frac{5}{119}](https://tex.z-dn.net/?f=%20e%5E%7B-0.12t%7D%20%3D%20%5Cfrac%7B%5Cfrac%7B5%7D%7B14%7D%7D%7B8.5%7D%3D%20%5Cfrac%7B5%7D%7B119%7D)
![ln e^{-0.12t} = ln (\frac{5}{119})](https://tex.z-dn.net/?f=%20ln%20e%5E%7B-0.12t%7D%20%3D%20ln%20%28%5Cfrac%7B5%7D%7B119%7D%29)
![-0.12 t = ln(\frac{5}{119})](https://tex.z-dn.net/?f=%20-0.12%20t%20%3D%20ln%28%5Cfrac%7B5%7D%7B119%7D%29)
![t = \frac{ln(\frac{5}{119})}{-0.12} = 26.414 weeks](https://tex.z-dn.net/?f=%20t%20%3D%20%5Cfrac%7Bln%28%5Cfrac%7B5%7D%7B119%7D%29%7D%7B-0.12%7D%20%3D%2026.414%20weeks)
d) If we find the limit when t tend to infinity for the function we have this:
![lim_{t \to \infty} \frac{95}{1+8.5 e^{-0.12t}} = 95](https://tex.z-dn.net/?f=%20lim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12t%7D%7D%20%3D%2095)
So then the number of words per minute have a limit and is 95 as t increases without bound.
Step-by-step explanation:
For this case we have the following expression for the average number of words per minutes typed adter t weeks:
![N(t) = \frac{95}{1+8.5 e^{-0.12t}}](https://tex.z-dn.net/?f=%20N%28t%29%20%3D%20%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12t%7D%7D)
Part a
For this case we just need to replace the value of t=10 in order to see what we got:
![N(t=10) = \frac{95}{1+8.5 e^{-0.12(10)}}= \frac{95}{1+ 8.5 e^{-1.2}} = 26.684](https://tex.z-dn.net/?f=%20N%28t%3D10%29%20%3D%20%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12%2810%29%7D%7D%3D%20%5Cfrac%7B95%7D%7B1%2B%208.5%20e%5E%7B-1.2%7D%7D%20%3D%2026.684)
So the number of words per minute typed after 10 weeks are approximately 27.
Part b
For this case we just need to replace the value of t=20 in order to see what we got:
![N(t=20) = \frac{95}{1+8.5 e^{-0.12(20)}}= \frac{95}{1+ 8.5 e^{-2.4}} = 53.639](https://tex.z-dn.net/?f=%20N%28t%3D20%29%20%3D%20%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12%2820%29%7D%7D%3D%20%5Cfrac%7B95%7D%7B1%2B%208.5%20e%5E%7B-2.4%7D%7D%20%3D%2053.639)
So the number of words per minute typed after 20 weeks are approximately 54.
Part c
For this case we want to solve the following equation:
![70 =\frac{95}{1+8.5 e^{-0.12t}}](https://tex.z-dn.net/?f=%2070%20%3D%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12t%7D%7D)
And we can rewrite this expression like this:
![1+ 8.5 e^{-0.12t} = \frac{95}{70}= \frac{19}{14}](https://tex.z-dn.net/?f=%201%2B%208.5%20e%5E%7B-0.12t%7D%20%3D%20%5Cfrac%7B95%7D%7B70%7D%3D%20%5Cfrac%7B19%7D%7B14%7D)
![8.5 e^{-0.12t} = \frac{19}{14}-1= \frac{5}{14}](https://tex.z-dn.net/?f=8.5%20e%5E%7B-0.12t%7D%20%3D%20%5Cfrac%7B19%7D%7B14%7D-1%3D%20%5Cfrac%7B5%7D%7B14%7D)
Now we can divide both sides by 8.5 and we got:
![e^{-0.12t} = \frac{\frac{5}{14}}{8.5}= \frac{5}{119}](https://tex.z-dn.net/?f=%20e%5E%7B-0.12t%7D%20%3D%20%5Cfrac%7B%5Cfrac%7B5%7D%7B14%7D%7D%7B8.5%7D%3D%20%5Cfrac%7B5%7D%7B119%7D)
Now we can apply natural log on both sides and we got:
![ln e^{-0.12t} = ln (\frac{5}{119})](https://tex.z-dn.net/?f=%20ln%20e%5E%7B-0.12t%7D%20%3D%20ln%20%28%5Cfrac%7B5%7D%7B119%7D%29)
![-0.12 t = ln(\frac{5}{119})](https://tex.z-dn.net/?f=%20-0.12%20t%20%3D%20ln%28%5Cfrac%7B5%7D%7B119%7D%29)
And then if we solve for t we got:
![t = \frac{ln(\frac{5}{119})}{-0.12} = 26.414 weeks](https://tex.z-dn.net/?f=%20t%20%3D%20%5Cfrac%7Bln%28%5Cfrac%7B5%7D%7B119%7D%29%7D%7B-0.12%7D%20%3D%2026.414%20weeks)
And we can see this on the plot 1 attached.
Part d
If we find the limit when t tend to infinity for the function we have this:
![lim_{t \to \infty} \frac{95}{1+8.5 e^{-0.12t}} = 95](https://tex.z-dn.net/?f=%20lim_%7Bt%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B95%7D%7B1%2B8.5%20e%5E%7B-0.12t%7D%7D%20%3D%2095)
So then the number of words per minute have a limit and is 95 as t increases without bound.