Answer:
a) For 2008 we have that t = 2008-2008 = 0 and we have:
![V(0)= 1.4e^{0.039*0}= 1.4](https://tex.z-dn.net/?f=%20V%280%29%3D%201.4e%5E%7B0.039%2A0%7D%3D%201.4)
For 2022 we have that t = 2022-2008=14 and if we replace we got:
![V(12) = 1.4 e^{0.039*14}=2.417](https://tex.z-dn.net/?f=%20V%2812%29%20%3D%201.4%20e%5E%7B0.039%2A14%7D%3D2.417)
b) ![2.8 = 1.4 e^{0.039 t}](https://tex.z-dn.net/?f=%202.8%20%3D%201.4%20e%5E%7B0.039%20t%7D)
We can divide both sides by 1.4 and we got:
![2 = e^{0.039 t}](https://tex.z-dn.net/?f=%202%20%3D%20e%5E%7B0.039%20t%7D)
Now natural log on both sides:
![ln (2) = 0.039 t](https://tex.z-dn.net/?f=%20ln%20%282%29%20%3D%200.039%20t)
![t = \frac{ln(2)}{0.039}= 17.77 years](https://tex.z-dn.net/?f=%20t%20%3D%20%5Cfrac%7Bln%282%29%7D%7B0.039%7D%3D%2017.77%20years)
Step-by-step explanation:
For this case we have the following model given:
![V(t) = 1.4 e^{0.039 t}](https://tex.z-dn.net/?f=%20V%28t%29%20%3D%201.4%20e%5E%7B0.039%20t%7D)
Where V represent the exports of goods and the the number of years after 2008.
Part a
Estimate the value of the country's exports in 2008 and 2022
For 2008 we have that t = 2008-2008 = 0 and we have:
![V(0)= 1.4e^{0.039*0}= 1.4](https://tex.z-dn.net/?f=%20V%280%29%3D%201.4e%5E%7B0.039%2A0%7D%3D%201.4)
For 2022 we have that t = 2022-2008=14 and if we replace we got:
![V(12) = 1.4 e^{0.039*14}=2.417](https://tex.z-dn.net/?f=%20V%2812%29%20%3D%201.4%20e%5E%7B0.039%2A14%7D%3D2.417)
Part b
What is the doubling time for the value of the country's exports.
For this case we can set up the following condition:
![2.8 = 1.4 e^{0.039 t}](https://tex.z-dn.net/?f=%202.8%20%3D%201.4%20e%5E%7B0.039%20t%7D)
We can divide both sides by 1.4 and we got:
![2 = e^{0.039 t}](https://tex.z-dn.net/?f=%202%20%3D%20e%5E%7B0.039%20t%7D)
Now natural log on both sides:
![ln (2) = 0.039 t](https://tex.z-dn.net/?f=%20ln%20%282%29%20%3D%200.039%20t)
![t = \frac{ln(2)}{0.039}= 17.77 years](https://tex.z-dn.net/?f=%20t%20%3D%20%5Cfrac%7Bln%282%29%7D%7B0.039%7D%3D%2017.77%20years)