The capital formation of the investment function over a given period is the
accumulated capital for the period.
- (a) The capital formation from the end of the second year to the end of the fifth year is approximately <u>298.87</u>.
- (b) The number of years before the capital stock exceeds $100,000 is approximately <u>46.15 years</u>.
Reasons:
(a) The given investment function is presented as follows;
![I(t) = 100 \cdot e^{0.1 \cdot t}](https://tex.z-dn.net/?f=I%28t%29%20%3D%20100%20%5Ccdot%20e%5E%7B0.1%20%5Ccdot%20t%7D)
(a) The capital formation is given as follows;
![\displaystyle Capital = \int\limits {100 \cdot e^{0.1 \cdot t}} \, dt =1000 \cdot e^{0.1 \cdot t}} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20Capital%20%3D%20%5Cint%5Climits%20%7B100%20%5Ccdot%20e%5E%7B0.1%20%5Ccdot%20t%7D%7D%20%5C%2C%20dt%20%3D1000%20%5Ccdot%20%20e%5E%7B0.1%20%5Ccdot%20t%7D%7D%20%2B%20C)
From the end of the second year to the end of the fifth year, we have;
The end of the second year can be taken as the beginning of the third year.
Therefore, for the three years; Year 3, year 4, and year 5, we have;
![\displaystyle Capital = \int\limits^5_3 {100 \cdot e^{0.1 \cdot t}} \, dt \approx 298.87](https://tex.z-dn.net/?f=%5Cdisplaystyle%20Capital%20%3D%20%5Cint%5Climits%5E5_3%20%7B100%20%5Ccdot%20e%5E%7B0.1%20%5Ccdot%20t%7D%7D%20%5C%2C%20dt%20%5Capprox%20298.87)
The capital formation from the end of the second year to the end of the fifth year, C ≈ 298.87
(b) When the capital stock exceeds $100,000, we have;
![\displaystyle \mathbf{\left[1000 \cdot e^{0.1 \cdot t}} + C \right]^t_0} = 100,000](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20%5Cmathbf%7B%5Cleft%5B1000%20%5Ccdot%20%20e%5E%7B0.1%20%5Ccdot%20t%7D%7D%20%2B%20C%20%5Cright%5D%5Et_0%7D%20%3D%20100%2C000)
Which gives;
![\displaystyle 1000 \cdot e^{0.1 \cdot t}} - 1000 = 100,000](https://tex.z-dn.net/?f=%5Cdisplaystyle%201000%20%5Ccdot%20%20e%5E%7B0.1%20%5Ccdot%20t%7D%7D%20-%201000%20%3D%20100%2C000)
![\displaystyle \mathbf{1000 \cdot e^{0.1 \cdot t}}} = 100,000 + 1000 = 101,000](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cmathbf%7B1000%20%5Ccdot%20%20e%5E%7B0.1%20%5Ccdot%20t%7D%7D%7D%20%3D%20100%2C000%20%2B%201000%20%3D%20101%2C000)
![\displaystyle e^{0.1 \cdot t}} = 101](https://tex.z-dn.net/?f=%5Cdisplaystyle%20e%5E%7B0.1%20%5Ccdot%20t%7D%7D%20%3D%20101)
![\displaystyle t = \frac{ln(101)}{0.1} \approx 46.15](https://tex.z-dn.net/?f=%5Cdisplaystyle%20t%20%3D%20%5Cfrac%7Bln%28101%29%7D%7B0.1%7D%20%5Capprox%2046.15)
The number of years before the capital stock exceeds $100,000 ≈ <u>46.15 years</u>.
Learn more investment function here:
brainly.com/question/25300925
Answer:Its B.
Step-by-step explanation:
If you take awa the sales tax it should be B. Sorry if it’s wrong tho.